- Math Article
- How To Solve Linear Differential Equation

## How to Solve Linear Differential Equation

A general first-order differential equation is given by the expression:

dy/dx + Py = Q where y is a function and dy/dx is a derivative.

The solution of the linear differential equation produces the value of variable y.

## Linear Differential Equations Definition

## Non-Linear Differential Equation

## Solving Linear Differential Equations

Multiplying both sides of equation (1) with the integrating factor M(x) we get;

M(x)dy/dx + M(x)Py = QM(x) …..(2)

Now we chose M(x) in such a way that the L.H.S of equation (2) becomes the derivative of y.M(x)

i.e. d(yM(x))/dx = (M(x))dy/dx + y (d(M(x)))dx … (Using d(uv)/dx = v(du/dx) + u(dv/dx)

⇒ M(x) /(dy/dx) + M(x)Py = M (x) dy/dx + y d(M(x))/dx

Integrating both sides with respect to x, we get;

\(\begin{array}{l} log M (x) = \int Pdx (As \int \frac {f'(x)}{f(x)} ) = log f(x) \end{array} \)

\(\begin{array}{l} M(x) = e^{\int Pdx}= I.F\end{array} \)

Multiplying both the sides of equation (1) by the I.F. we get

\(\begin{array}{l} e^{\int Pdx}\frac{dy}{dx} + yPe^{\int Pdx} = Qe^{\int Pdx} \end{array} \)

This could be easily rewritten as:

Now integrating both the sides with respect to x, we get:

\(\begin{array}{l} \int d(y.e^{\int Pdx }) = \int Qe^{\int Pdx}dx + c \end{array} \)

\(\begin{array}{l} y = \frac {1}{e^{\int Pdx}} (\int Qe^{\int Pdx}dx + c )\end{array} \)

where C is some arbitrary constant.

## How to Solve First Order Linear Differential Equation

Learn to solve the first-order differential equation with the help of steps given below.

where P and Q are constants or functions of the independent variable x only.

\(\begin{array}{l} e^{\int Pdx} = I.F\end{array} \)

\(\begin{array}{l} e^{\int Pdx} \frac{dy}{dx} + yPe^{\int Pdx} = Qe^{\int Pdx} \end{array} \)

where C is some arbitrary constant

## Solved Examples

Example 1: Solve the LDE = dy/dx = [1/(1+x 3 )] – [3x 2 /(1 + x 2 )]y

The above mentioned equation can be rewritten as dy/dx + [3x 2 /(1 + x 2 )] y = 1/(1+x 3 )

Comparing it with dy/dx + Py = O , we get

Let’s figure out the integrating factor(I.F.) which is,

\(\begin{array}{l} e^{\int Pdx} \end{array} \)

\(\begin{array}{l}I.F = e^{\int \frac {3x^2}{1 + x^3}} dx = e^{ln (1 + x^3)} \end{array} \)

Now, we can also rewrite the L.H.S as:

⇒ d(y × (1 + x 3 )) dx = [1/(1 +x 3 )] × (1 + x 3 )

Integrating both the sides w. r. t. x, we get,

Solve the following differential equation:

Comparing the given equation with dy/dx + Py = Q

Now lets find out the integrating factor using the formula

\(\begin{array}{l} e^{\int Pdx}= I.F \end{array} \)

\(\begin{array}{l} e^{\int secdx}= I.F. \end{array} \)

\(\begin{array}{l} I.F. = e^{ln |sec x + tan x |} = sec x + tan x \end{array} \)

Now we can also rewrite the L.H.S as

⇒ d(y × (sec x + tan x ))/dx = 7(sec x + tan x)

\(\begin{array}{l} \int d ( y × (sec x + tan x )) = \int 7(sec x + tan x) dx \end{array} \)

\(\begin{array}{l} y =\frac {7(ln|sec x + tan x| + log|sec x| }{(sec x + tan x)} + c \end{array} \)

We know that the slope of the tangent at (x,y) is,

tanƟ= dy/dx = (x 4 + 2xy + 1)/1 – x 2

Reframing the equation in the form dy/dx + Py = Q , we get

dy/dx = 2xy/(1 – x 2 ) + (x 4 + 1)/(1 – x 2 )

⇒ dy/dx – 2xy/(1 – x 2 ) = (x 4 + 1)/(1 – x 2 )

Comparing we get P = -2x/(1 – x 2 )

Now, let’s find out the integrating factor using the formula.

\(\begin{array}{l} e^{\int \frac{-2x}{1-x^2}}dx = e^{ln (1 – x^2)} = 1 – x^2 =I.F \end{array} \)

\(\begin{array}{l} \frac {d(y × I.F)}{dx}, \end{array} \)

Integrating both sides w. r. t. x, we get,

\(\begin{array}{l} \Rightarrow y × (1 – x^2) = \int x^4 + 1 dx …(1) \end{array} \)

⇒ y = x 5 /5 + x/(1 – x 2 ) + C

## Frequently Asked Questions – FAQs

Your Mobile number and Email id will not be published. Required fields are marked *

## Register with BYJU'S & Download Free PDFs

- Practice and Assignment problems are not yet written. As time permits I am working on them, however I don't have the amount of free time that I used to so it will take a while before anything shows up here.
- Show all Solutions/Steps/ etc.
- Hide all Solutions/Steps/ etc.
- First Order DE's Introduction
- Separable Equations
- Basic Concepts
- Second Order DE's
- Calculus II
- Calculus III
- Differential Equations
- Algebra & Trig Review
- Common Math Errors
- Complex Number Primer
- How To Study Math
- Cheat Sheets & Tables
- MathJax Help and Configuration
- Notes Downloads
- Complete Book
- Practice Problems Downloads
- Problems not yet written.
- Assignment Problems Downloads
- Other Items
- Get URL's for Download Items
- Print Page in Current Form (Default)
- Show all Solutions/Steps and Print Page
- Hide all Solutions/Steps and Print Page

## Section 2.1 : Linear Differential Equations

The final step is then some algebra to solve for the solution, \(y(t)\).

Divide both sides by \(\mu \left( t \right)\),

As with the process above all we need to do is integrate both sides to get.

Exponentiate both sides to get \(\mu \left( t \right)\) out of the natural logarithm.

The solution to a linear first order differential equation is then

## Solution Process

The solution process for a first order linear differential equation is as follows.

- Put the differential equation in the correct initial form, \(\eqref{eq:eq1}\).
- Find the integrating factor, \(\mu \left( t \right)\), using \(\eqref{eq:eq10}\).
- Multiply everything in the differential equation by \(\mu \left( t \right)\) and verify that the left side becomes the product rule \(\left( {\mu \left( t \right)y\left( t \right)} \right)'\) and write it as such.
- Integrate both sides, make sure you properly deal with the constant of integration.
- Solve for the solution \(y(t)\).

First, we need to get the differential equation in the correct form.

From this we can see that \(p(t)=0.196\) and so \(\mu \left( t \right)\) is then.

Okay. It’s time to play with constants again. We can subtract \(k\) from both sides to get.

So, it looks like we did pretty good sketching the graphs back in the direction field section.

So, the actual solution to the IVP is.

A graph of this solution can be seen in the figure above.

Let’s do a couple of examples that are a little more involved.

Rewrite the differential equation to get the coefficient of the derivative to be one.

Now find the integrating factor.

Also note that we made use of the following fact.

Finally, apply the initial condition to find the value of \(c\).

Below is a plot of the solution.

First, divide through by the t to get the differential equation into the correct form.

Now let’s get the integrating factor, \(\mu \left( t \right)\).

and rewrite the integrating factor in a form that will allow us to simplify it.

Integrate both sides and solve for the solution.

Finally, apply the initial condition to get the value of \(c\).

Here is a plot of the solution.

First, divide through by \(t\) to get the differential equation in the correct form.

Now that we have done this we can find the integrating factor, \(\mu \left( t \right)\).

Now, we just need to simplify this as we did in the previous example.

Again, we can drop the absolute value bars since we are squaring the term.

Apply the initial condition to find the value of \(c\).

First, divide through by a 2 to get the differential equation in the correct form.

Now find \(\mu \left( t \right)\).

This behavior can also be seen in the following graph of several of the solutions.

## Linear Differential Equation

## What Is a Linear Differential Equation?

## Derivation for Solution of Linear Differential Equation

g(x).dy/dx + P.g(x).y = Q.g(x)

Choose g(x) in such a way such that the RHS becomes the derivative of y.g(x).

g(x).dy/dx + P.g(x)y = d/dx(y.g(x)]

g(x).dy/dx + P.g(x).y = g(x).dy/dx + y.g'(x)

Integrating both sides with respect to x, we get

\(\int P.dx= \int \frac{g'(x)}{g(x)}.dx\)

\(e^{\int P.dx}.\dfrac{dy}{dx} + Pe^{\int P.dx}y = Q.e^{\int P.dx}\)

\(\dfrac{d}{dx}(y.e^{\int P.dx} )= Qe^{\int P.dx}\)

Integrating both sides, with respect to x the following expression is obtained..

\(y.e^{\int P.dx} =\int (Q.e^{\int P.dx}.dx)\)

\(y=e^{-\int P.dx} .\int (Q.e^{\int P.dx}.dx) + C\)

The above expression is the general solution of the linear differential eqution.

## Formula for General Solution of Linear Differential Equation

- The general solution of the differential equation dy/x +Py = Q is as follows. \(y.(I.F)=\int (Q.(I.F).dx)+ C\). Here we have Integrating Factor (I.F) = \(e^{\int P.dx}\).
- Also the general solution of the differential equation dx/y +Px = Q is as follows. \(x.(I.F)=\int (Q.(I.F).dy)+ C\). Here we have Integrating Factor (I.F) = \(e^{\int P.dy}\).

## Steps to Solve Linear Differential Equation

- Step - I: Simplify and write the given differential equation in the form dy/dx + Py = Q, where P and Q are numeric constants or functions in x.
- Step - II: Find the Integrating Factor of the linear differential equation (IF) = \(e^{\int P.dx}\).
- Step-III: Now we can write the solution of the linear differential equation as follows. \(y(I.F) = \int(Q × I.F).dx + C\)

- Integration Formulas
- Differentiation and Integration Formulas
- Chain Rule Formula
- Differential Equations

## Examples on Linear Differential Equation

Example 1: Find the general solution of the differential equation xdy -(y + 2x 2 ).dx = 0

Further, the solution of the differential equation is as follows.

Example 2: Find the derivative of dy/dx + Secx.y = Tanx

Further the integrating factor is I.F = \(e^{\int Secx.dx}

## Practice Questions on Linear Differential Equation

Faqs on linear differential equation.

## How Do You Know If a Differential Equation Is a Linear Differential Equation?

## How Do You Solve a Linear Differential Equation?

## What Is the Standard Form of Linear Differential Equation in x?

## What Is the Formula For the General Solution of Linear Differential Equation

## Stack Exchange Network

Connect and share knowledge within a single location that is structured and easy to search.

## Linear vs nonlinear differential equation

- 9 $\begingroup$ linear equations must involve $y, y', y''$ etc. with coefficients that are (at worst) functions of $x$. terms like $yy'$ or $y^2$ are ruled out $\endgroup$ – citedcorpse Jun 8, 2013 at 10:10
- $\begingroup$ If the ODE has the unknown function and/or its derivative(s) as an argument of a trigonometric, hyperbolic trigonometric, exponential, logarithmic, and/or n-th root function, the ODE is non-linear. If the ODE has a product of the unknown function times any of its derivatives, the ODE is non-linear. If the ODE has the unknown function and/or its derivative(s) with power greater than 1, the ODE is non-linear. $\endgroup$ – alejnavab Feb 2, 2022 at 5:46

## 5 Answers 5

Your first case is indeed linear, since it can be written as:

$$\left(\frac{d^2}{dx^2} - 2\right)y = \ln(x)$$

While the second one is not. To see this first we regroup all $y$ to one side:

- $\begingroup$ does this mean that linear differential equation has one y, and non-linear has two y, y'? $\endgroup$ – maycca Jun 21, 2017 at 8:28
- $\begingroup$ @Daniel Robert-Nicoud does the same thing apply for linear PDE? Wikipedia says PDE is linear if it is linear in dependent variable and its derivatives. $\endgroup$ – ramanujan Nov 20, 2018 at 20:20
- 1 $\begingroup$ @ramanujan Yes, it does. $\endgroup$ – Daniel Robert-Nicoud Nov 20, 2018 at 21:55

Each coefficient depends only on the independent variable $x$.

Always go back to the definitions. :-)

- 1 $\begingroup$ your statement is true for only homogeneous LDE? For nonhomogeneous it is false. And I think it is also false in PDE. $\endgroup$ – ramanujan Nov 20, 2018 at 20:16
- $\begingroup$ @ramanujan What is an example of a non-homogeneous LDE or PDE whose solutions' linear combinations are also solutions? $\endgroup$ – Geremia Nov 20, 2018 at 22:29
- $\begingroup$ (do you mean 'solution of linear combinations are not solutions?') Simplest example is $y^{\prime}= 2$ and take both solutions $y=2 x$. $\endgroup$ – ramanujan Nov 21, 2018 at 4:21
- $\begingroup$ @ramanujan $y'=2$ is non-homogeneous? $\endgroup$ – Geremia Nov 21, 2018 at 4:37
- $\begingroup$ Cited sources says first order LDE is nonhomogeneous if $y^{\prime} + p(x)y = q(x)$, if $q(x)$ is not identically zero. here and here $\endgroup$ – ramanujan Nov 21, 2018 at 7:38

Because highest order derivative is multiplied with dependent variable $y$. Like $y y'$.

## You must log in to answer this question.

Not the answer you're looking for browse other questions tagged ordinary-differential-equations ..

## Hot Network Questions

- FAA Handbooks Copyrights
- get unique without sorting in jq
- Disconnect between goals and daily tasks...Is it me, or the industry?
- How can I check before my flight that the cloud separation requirements in VFR flight rules are met?
- Do "superinfinite" sets exist?
- What did Ctrl+NumLock do?
- Are the plants animated by an Assassin Vine considered magical?
- Conjugation of the Auxiliary Verb 得る When it's Read as うる
- How to tell which packages are held back due to phased updates
- What is the point of Thrower's Bandolier?
- What is temperature in the classical entropy definition?
- A story about a girl and a mechanical girl with a tattoo travelling on a train
- How can we prove that the supernatural or paranormal doesn't exist?
- How do you ensure that a red herring doesn't violate Chekhov's gun?
- Why did Windows 3.0 fail in Japan?
- Copyright issues when journal is defunct
- AC Op-amp integrator with DC Gain Control in LTspice
- Who owns code in a GitHub organization?
- Drywall repair - trying to match texture
- Haunted house movie that focuses on a basement door and a ghost who wants to steal a mother's child
- How should I go about getting parts for this bike?
- Do new devs get fired if they can't solve a certain bug?
- Does high pressure reverse reaction between zinc and sulfuric acid?
- Why do many companies reject expired SSL certificates as bugs in bug bounties?

## Linear differential equation problems and answers

## Linear Differential Equation

5.3 first order linear differential equations.

## Linear Differential Equation (Solution & Solved Examples)

Get calculation support online

Looking for a little help with your math homework? Check out our online calculation assistance tool!

Math understanding that gets you

If you want to improve your math skills, the best way is to practice as often as possible.

If you want to get the best homework answers, you need to ask the right questions.

You can save time by doing things more efficiently.

Work on the task that is interesting to you

You can work on whatever task interests you the most.

## Linear Differential Equation

## Linear Differential Equation Formula

## Linear Differential Equation Properties

The linear differential equations have the following properties.

a] The y function and its respective derivatives come in the equation till the first degree only.

b] The products of y and/or any of its respective derivatives are not present.

c] No functions that are transcendental.

## Linear Differential Equation Examples

Example 1: Solve the equation: x \frac{d y}{d x}-2 y=x^{3} \cos 4 x .

Obtain the values of P and Q by comparing it to the standard form of linear differential equation.

P = (- 2 / x) and Q = x^{2} \cos 4 x

Compute the integrating factor by using the below formula.

Example 2: Solve the equation: y’ (x) + y (x) / x = 3x.

Example 3: Solve the equation: y^{\prime}-y-x e^{x}=0

Example 4: Solve the differential equation xy'=y+2x^{3}

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

## Differential equations

First order differential equations, second order linear equations, laplace transform.

## with Answer, Solution, Formula - Homogeneous Differential Equations: Solved Example Problems | 12th Business Maths and Statistics : Chapter 4 : Differential Equations

Solve the differential equation y 2 dx + ( xy + x 2 ) dy = 0

Given marginal cost function is (x 2 + xy) dy + (3xy + y 2 )dx=0

Privacy Policy , Terms and Conditions , DMCA Policy and Compliant

Copyright © 2018-2023 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.

## Select your language

## Initial Value Problem Differential Equations

- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Area Between Two Curves
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits at Infinity
- Limits at Infinity and Asymptotes
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Radius of Convergence
- Related Rates
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Second Derivative Test
- Separable Equations
- Separation of Variables
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Dynamic Programming
- Formulating Linear Programming Problems
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Reflection in Geometry
- Regular Polygon
- Right Triangles
- SSS and SAS
- Segment Length
- Similarity Transformations
- Special quadrilaterals
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Translations
- Triangle Inequalities
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Conservation of Mechanical Energy
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Elastic Strings and Springs
- Force as a Vector
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Work Done by a Constant Force
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Argand Diagram
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Circle Theorems
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- De Moivre's Theorem
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integrating Polynomials
- Integrating Trigonometric Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radical Functions
- Rates of Change
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences and Series
- Series Maths
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Variables in Algebra
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Distribution Function
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Degrees of Freedom
- Discrete Random Variable
- Distributions
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Hypothesis Test for Correlation
- Hypothesis Test for Regression Slope
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Paired T-Test
- Point Estimation
- Probability
- Probability Calculations
- Probability Density Function
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Random Variables
- Randomized Block Design
- Residual Sum of Squares
- Sample Mean
- Sample Proportion
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Spearman's Rank Correlation Coefficient
- Standard Deviation
- Standard Error
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- T-distribution
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Variance for Binomial Distribution
- Venn Diagrams

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Nie wieder prokastinieren mit unseren Lernerinnerungen.

## Definition of an Initial-Value Problem

\[ \begin{align} &y' = f(x,y) \\ &y(a) = b \end{align} \]

where \((a,b)\) is the point the solution \(y(x)\) must go through.

## First Order Differential Equation Initial Value Problem

Let's start with the constant coefficient first order linear differential equation

\[y(x) =Ce^{-Ax}+\frac{B}{A}\]

\[ y_1=Ce^{-Ax_1}+\frac{B}{A}, \]

\[ C = \frac{y_1 - \frac{B}{A}}{e^{-Ax_1}}. \]

That means the solution to the IVP

\[ \begin{align} y' &= f(x,y) \\ y(x_1) &= y_1 \end{align} \]

\[ \begin{align} &y' +3y = 7 \\ &y(0) = \frac{1}{3}. \end{align} \]

The first step is finding the general solution, which for this problem is

\[y(x) =Ce^{-3x}+\frac{7}{3}.\]

\[\frac{1}{3} =Ce^{-3\cdot 0}+\frac{7}{3},\]

so \(C = -2\). That means the solution to the IVP is

\[y(x) = -2e^{-3x}+\frac{7}{3}.\]

\[ y(x) = \left(y_1 - \frac{B}{A} \right)e^{-A(x-x_1)} +\frac{B}{A}.\]

Solutions exist on the whole real line.

Now let's move on to the first order linear differential equation

\[ \alpha(x)=e^{\int P(x)\,\mathrm{d}x},\]

\[ y(x) = \frac{1}{\alpha (x)} \left( \int \alpha(x) \, Q(x) \, \mathrm{d}x + C \right).\]

Finding the solution to the IVP

\[ \begin{align} y'+P(x)y&=Q(x) \\ y(a) &= b \end{align} \]

\[ \begin{align} &y'+\frac{1}{x}y=\frac{1}{\sqrt{x}} \\ &y(1) = \frac{5}{3} .\end{align} \]

First let's find the general solution. The integrating factor is

Then the general solution is given by

Now you can plug in the initial value \( y(1) = \dfrac{5}{3} \) to get

\[ \frac{5}{3} = \frac{2\sqrt{1}}{3} + \frac{C}{1}, \]

\[C = \frac{5}{3} - \frac{2}{3} = 1.\]

With the assumption that \(x>0\), the solution to the IVP is

\[ y(x) = \frac{2\sqrt{x}}{3} + \frac{1}{x}.\]

\[\begin{align} &y' + P(x)y = Q(x) \\ &y(a) = b \end{align}\]

exists and is unique on the whole real line .

## Initial value Problems and Separable Differential Equations

Remember that a differential equation is separable if you can write it in the form

\[\begin{align} &N(y)y' = M(x) \\ &y(a) = b. \end{align}\]

Let's take a look at an example.

\[ \begin{align} &y' = \frac{y^2}{x} \\ &y(0) = 4. \end{align} \]

First, you can rewrite the differential equation as

\[ \frac{1}{y^2} y' = \frac{1}{x}, \]

so it is a separable differential equation. Separating variables and integrating gives you

\[ \int \frac{1}{y^2} \, \mathrm{d}y = \int \frac{1}{x} \, \mathrm{d}x\]

\[ -\frac{1}{y} = \ln |x| + C.\]

One more example, to see the kinds of things that can happen.

\[ \begin{align} &y' = y^{\frac{1}{3}} \\ &y(0) = 0. \end{align} \]

What about other solutions? This is a separable equation, so separating and integrating gives you

\[ \int \frac{1}{y^\frac{1}{3} } \, \mathrm{d}y = \int 1\, \mathrm{d}x\]

\[ \frac{3}{2}y^\frac{2}{3} = x+ C.\]

Using the initial value \(y(0) = 0\) to find \(C\), you get

\[ \frac{3}{2}0^\frac{2}{3} = 0+ C,\]

so \(C=0\). That means there is a second solution to this IVP, namely the implicit solution

\[ \frac{3}{2}y^\frac{2}{3} = x.\]

You can get the explicit solution by solving for \(y\). If you do, you get

\[ y^\frac{2}{3} = \frac{2}{3}x,\]

\[ y(x) =\left( \frac{2}{3}x \right)^{\frac{3}{2}}.\]

More examples are always good!

## Examples of Differential Equation Initial Value Problems

Let's look more examples involving IVPs.

\[ \begin{align} &2xy'+4y=3 \\ &y(2) = \frac{5}{4} \end{align} \]

by first showing that the general solution is

\[y(x) = \frac{C}{x^2} + \frac{3}{4}.\]

What is the interval of existence for the solution to the IVP?

\[ y'(x) = \frac{-2C}{x^3} .\]

Now to solve the IVP. Using the initial value,

\[ y(2) = \frac{C}{2^2} + \frac{3}{4} = \frac{5}{4}. \]

Solving for \(C\), you can see that \(C=2\). So the solution to the IVP is

\[y(x) = \frac{2}{x^2} + \frac{3}{4}.\]

Sometimes you will have a solution which is implicit, and that can be used to solve an IVP as well.

an implicit solution to the IVP

\[ \begin{align} &y' = \frac{x}{y} \\ &y(2) = 1 ?\end{align} \]

Then using implicit differentiation and remembering that \(y\) is a function of \(x\),

Looking first at the left hand side, and using the fact that if it is a solution then

\[ \frac{\mathrm{d}}{\mathrm{d} x} \left(x^2 - 3 \right) = 2x.\]

Since both sides are the same, \(y(x)\) does satisfy the differential equation.

Therefore the proposed implicit solution does actually solve the IVP.

What do you do if you can't find an implicit or explicit solution to an IVP?

## Numerical Initial Value Problems in Ordinary Differential Equations

## Initial Value Problem Differential Equations - Key takeaways

- In differential equations, initial value problem is often abbreviated IVP.
- An IVP is a differential equation together with a place for a solution to start, called the initial value.
- IVPs are often written\[ \begin{align} &y' = f(x,y) \\ &y(a) = b \end{align} \] where \((a,b)\) is the point the solution \(y(x)\) must go through.
- When solving separable IVPs it is important to choose the interval of existence that contains the initial value.
- Not all IVPs have a unique solution, or even have a solution!

## Frequently Asked Questions about Initial Value Problem Differential Equations

--> how to solve initial value problems in differential equations .

## --> What is initial value problem in differential equation?

It is a differential equation where the solution is required to go through a specific point.

## --> What is a initial value example?

## --> What is the initial value in an equation?

--> does the initial value problem have a unique solution .

## Final Initial Value Problem Differential Equations Quiz

What is the abbreviation for an initial value problem in differential equations?

In terms of the solution of a differential equation, what does the initial value tell you?

It gives you a point the solution must go through.

What is the notation for an IVP?

\[ \begin{align} y' &= f(x,y) \\ y(a) &= b \end{align} \]

What does a first order linear constant coefficient IVP look like?

\[ \begin{align} y'+Ay&=B \\ y(x_1) &= y_1, \end{align}\]

where \(A, B, x_1\), and \(y_1\) are real numbers with \(A \neq 0\).

Name 4 properties of first order linear constant coefficient IVPs.

Solutions to the IVP are unique .

Solutions all have the same long term behavior.

First order linear constant coefficient initial value problems.

When does a first order linear IVP have a unique solution on the whole real line?

\[\begin{align} y' + P(x)y &= Q(x) \\ y(a) &= b \end{align}\]

What does a separable differential equation IVP look like?

So for real numbers \(a\) and \(b\), the IVP is

\[\begin{align} N(y)y' &= M(x) \\ y(a) &= b. \end{align}\]

What do you have to look out for when solving separable IVPs?

You need to be sure to pick the interval of existence that contains the initial value.

What can you do if you can't find an explicit or implicit solution to an IVP?

Try a numerical method, like Euler's Method.

What is Euler's Method used for?

Finding a numerical approximation to the solution of an IVP.

No. Some don't have solutions at all.

Do IVPs all have unique solutions?

No, some IVPs have two, or sometimes even more, solutions!

Will you always be able to find an explicit solution to an IVP, assuming there is a solution?

No. Sometimes all you will be able to find is an implicit solution.

## More explanations about Calculus

Be perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

## Study Analytics

Identify your study strength and weaknesses.

## Weekly Goals

Set individual study goals and earn points reaching them.

## Smart Reminders

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

## Magic Marker

Create flashcards in notes completely automatically.

## Smart Formatting

Create the most beautiful study materials using our templates.

## Join millions of people in learning anywhere, anytime - every day

Sign up to highlight and take notes. It’s 100% free.

## This is still free to read, it's not a paywall.

You need to register to keep reading, get free access to all of our study material, tailor-made.

Over 10 million students from across the world are already learning smarter.

## StudySmarter bietet alles, was du für deinen Lernerfolg brauchst - in einer App!

Tom M. Apostol (Paperback - Nov …

Anton, Bivens, Davis (Paperback …

Differential Calculus for IIT-JEE

Amit M Agarwal (Paperback - Jul 4…

Calculus and Analytic Geometry

Problems in Calculus of One Variable...

Best Price $76.00 or Buy New $134.49

Best Price $150.00 or Buy New $197.98

Best Price $94.51 or Buy New $165.24

## Introduction to Differential Equations

Sign in | Recent Site Activity | Report Abuse | Print Page | Powered By Google Sites

## Math Insight

Examples of solving linear ordinary differential equations using an integrating factor.

## Thread navigation

- Previous: Solving linear ordinary differential equations using an integrating factor
- Next: Online quiz: Scalar linear equation problems

## Math 5447, Fall 2022

- Solving linear ordinary differential equations using an integrating factor
- An introduction to ordinary differential equations
- Ordinary differential equation examples
- Exponential growth and decay: a differential equation
- Another differential equation: projectile motion
- Solving single autonomous differential equations using graphical methods
- Spruce budworm outbreak model
- Single autonomous differential equation problems
- Introduction to visualizing differential equation solutions in the phase plane
- Two dimensional autonomous differential equation problems
- More similar pages

## Slope in equation form

## What is Slope Formula? Equation, Examples

If you're looking for a punctual person, you can always count on me.

Work on the task that is interesting to you

## Finding the Slope of a Linear Equation

If you need help, our customer support team is available 24/7 to assist you.

You can always count on our 24/7 customer support to be there for you when you need it.

Expert instructors will give you an answer in real-time

If you want to get things done, you need to set some deadlines.

## Find csc x if sin x + cot x cos x = .

In this blog post, we will take a look at how to Find csc x if sin x + cot x cos x = ..

## How do users think about us

Lifesaver. Definitely worth downloading, life-saver for many of my classmates. Also add "to the power" option in calculator section, i take a picture and look for the results, gREAT GREAT GREAT APP IT WORKS I WORKED ON A EXAM AND I GOT 100%.

This is the only thing between me and failing algebra 2. Best app I've tried for helping understand college level math, highly recommend it, help's with math SO much, it can solve any mathematical problem. I salute you developers. Best app for math homework.

Be sure to learn. Thank you so mcfreaking much. It's the reason I'm even passing. I feel like I'm cheating the system 🤫. This app works really well,but it doesn't always show you the answer, maybe we could see an ad in exchange for seeing them. Hands down, best app I have on my phone.

## Verify the Identity sin(x)+cos(x)cot(x)=csc(x)

Because the two sides have been shown to be equivalent, the equation is an identity. csc(x)

## TRIGONOMETRY

Find csc x if sin x + cot x cos x = 3..

At 24/7 Customer Help, we're always here to help you with your questions and concerns.

If you want to improve your theoretical performance, you need to put in the work.

## What is considered depreciation

What is considered depreciation is a mathematical tool that helps to solve math equations.

## Why people love us

## What Can Be Depreciated in Business? Depreciation Decoded

## What Is Depreciation, and How Is It Calculated?

Depreciation & how it affects your business.

Mathematics is a way of dealing with tasks that require e#xact and precise solutions.

Math can be difficult, but with a little practice, it can be easy!

Doing math equations is a great way to keep your mind sharp and improve your problem-solving skills.

Figure out mathematic equation

If you want to get the best homework answers, you need to ask the right questions.

Get the Most useful Homework explanation

## Topic No. 704 Depreciation

Have more time for your pursuits

You can have more time for your pursuits by learning to manage your time more efficiently.

## What is the math app that gives you answers

## The 7 Best Android Apps to Help You Solve Math Problems

Math is a way of solving problems by using numbers and equations.

If you're looking for a punctual person, you can always count on me.

## What people are saying about us

## Mathway: Math Problem Solver 4+

## Difference between fx991es and fx991ex

## Comparison Chart

A: The fx-991EX CLASSWIZ is better than the fx-115ES

## Do you know the difference between the Casio FX

## Which scientific calculator is best for Engineers? (2021)

Not bad at all! You just put your camera towards your problem and it solves it! With all possible explanations and processes, the scanner works very well even reading my slanted writing, this app has held my math education together for many years.

I am able to calculate all type of problems and thank you for adding the feature of showing calculations step by step, thank you for your good work done âœ…. Because first and foremost it helps me calculate fractions and expressions well.

This definitely motivated me to solve math problems on my own because I could just check if its right or wrong Š i wont abuse it because if face to face classes would start again i would definitely suffer if I dont know how to solve† P.

Math can be difficult, but with a little practice, it can be easy!

## 90 day fiance other way tell all 2020

## Why Was There No '90 Day Fiance: The Other Way' Season 2

## 90 Day Fianc Tell All Part 2 (TV Episode 2020)

When is '90 day fiance: the other way' tell.

Doing math equations is a great way to keep your mind sharp and improve your problem-solving skills.

Solve step-by-step equations to find the value of the variable.

Improve your theoretical performance

The best way to improve your theoretical performance is to practice as often as possible.

Our team of top experts are here to help you with all your needs.

## 90 Day Fianc: The Other Way (TV Series 2019

Math is a way of understanding the world around us.

If you want to enhance your academic performance, you need to be willing to put in the work.

## '90 Day Fianc: The Other Way': Which Couples Are Still

Math can be tough to wrap your head around, but with a little practice, it can be a breeze!

Our team is available 24/7 to help you with whatever you need.

Get detailed step-by-step explanations

Looking for detailed, step-by-step answers? Look no further – our experts are here to help.

Solving math problems can be a fun and rewarding experience.

Math is the study of numbers, shapes, and patterns.

## 90 Day Fianc: The Other Way

The best way to protect your data is to keep it secure.

Free time to spend with your friends

I love spending time with my friends when I have free time.

## IMAGES

## VIDEO

## COMMENTS

Consider the differential equation x ″ + 7x ′ + 12x = 0. Both e − 3t and 2e − 3t are solutions (you can check this). However, x(t) = c1e − 3t + c2(2e − 3t) is not the general solution. This expression does not account for all solutions to the differential equation.

We work a couple of examples of solving differential equations involving Dirac Delta functions and unlike problems with Heaviside functions our only real option for this kind of differential equation is to use Laplace transforms. We also give a nice relationship between Heaviside and Dirac Delta functions.

Examples of linear differential equations are: xdy/dx+2y = x 2 dx/dy - x/y = 2y dy/dx + ycot x = 2x 2 How to solve the first order differential equation? First write the equation in the form of dy/dx+Py = Q, where P and Q are constants of x only Find integrating factor, IF = e ∫Pdx Now write the solution in the form of y (I.F) = ∫Q × I.F C

The solution to a linear first order differential equation is then y(t) = ∫ μ(t)g(t) dt +c μ(t) (9) (9) y ( t) = ∫ μ ( t) g ( t) d t + c μ ( t) where, μ(t) = e∫p(t)dt (10) (10) μ ( t) = e ∫ p ( t) d t Now, the reality is that (9) (9) is not as useful as it may seem.

Examples on Linear Differential Equation Example 1: Find the general solution of the differential equation xdy - (y + 2x 2 ).dx = 0 Solution: The give differential equation is xdy - (y + 2x 2 ).dx = 0. This can be simplified to represent the following linear differential equation. dy/dx - y/x = 2x

$\begingroup$ If the ODE has the unknown function and/or its derivative(s) as an argument of a trigonometric, hyperbolic trigonometric, exponential, logarithmic, and/or n-th root function, the ODE is non-linear. If the ODE has a product of the unknown function times any of its derivatives, the ODE is non-linear. If the ODE has the unknown function and/or its derivative(s) with power greater ...

Linear differential equation problems and answers - We'll provide some tips to help you select the best Linear differential equation problems and answers for ... Examples on Linear Differential Equation Example 1: Find the general solution of the differential equation xdy -(y + 2x2).dx = 0. Solution: The give. Average satisfaction rating 4.7/5 ...

Linear Differential Equation Examples Example 1: Solve the equation: x \frac {d y} {d x}-2 y=x^ {3} \cos 4 x xdxdy − 2y = x3 cos4x. Answer: Convert the given equation into the standard form (dy / dx) + Py = Q of the linear differential equation. \frac {d y} {d x}-\frac {2} {x} y=x^ {2} \cos 4 x dxdy − x2y = x2 cos4x

Maths: Differential Equations: Linear differential equations of first order : Solved Example Problems with Answer, Solution, Formula Example Example 4.24 A firm has found that the cost C of producing x tons of certain product by the equation x dC/dx = 3/x − C and C = 2 when x = 1. Find the relationship between C and x. Solution: Prev Page Next Page

With numbers, you could think of this equation as ax + 5 = 5, where you control the variable a, but the variable x is outside your control, and can be any number whatsoever. The only way you can make this equation true is by making the variable a (the one you control), equal to 0, that way the variable that is outside your control stops messing the equation.

Learn differential equations for free—differential equations, separable equations, exact equations, integrating factors, and homogeneous equations, and more. ... Second order linear equations Complex and repeated roots of characteristic equation: Second order linear equations Method of undetermined coefficients: ...

Therefore, the solution to the initial-value problem is EXAMPLE 3 Solve . SOLUTION The given equation is in the standard form for a linear equation. Multiplying by the integrating factor ... 4 LINEAR DIFFERENTIAL EQUATIONS EXAMPLE 4 Suppose that in the simple circuit of Figure 4 the resistance is and the

In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small.It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution.

Example 4.18. If the marginal cost of producing x shoes is given by (3xy + y2 ) dx + (x 2 + xy) dy = 0 and the total cost of producing a pair of shoes is given by ₹12. Then find the total cost function.

In differential equations, initial value problem is often abbreviated IVP. An IVP is a differential equation together with a place for a solution to start, called the initial value. IVPs are often written y ′ = f ( x, y) y ( a) = b where ( a, b) is the point the solution y ( x) must go through.

A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant coefficients.

A diﬀerential equation. dy/dx = f1 (x, y) / f2 (x, y) where f1 and f2 are homogeneous functions of x and y of the same degree, is called a homogeneous diﬀerential equation. Such equations can be solved by taking a new dependent variable v connected with the old one y by the equation y = vx.

Next: Online quiz: Scalar linear equation problems; Similar pages. Solving linear ordinary differential equations using an integrating factor; An introduction to ordinary differential equations; Ordinary differential equation examples; Exponential growth and decay: a differential equation; Another differential equation: projectile motion

What is an nonlinear ordinary differential equation and give an. An equation in which the maximum degree of a term is 2 or more than two is called a nonlinear equation. + 2x + 1 = 0, 3x + 4y = 5, this is the example of. order now.

Nonhomogeneous differential equation examples - We'll provide some tips to help you select the best Nonhomogeneous differential equation examples for your ... Explain mathematic problems. Mathematics is the study of numbers, shapes, and patterns. It is used to solve problems. ... NonHomogeneous Second Order Linear Equations (Section 17.2 ...

Order and Degree of Differential Equations with Examples. Linear differential equations. The general linear ODE of order n is. (1) think of the formal polynomial p(D) as operating on a function y(x), converting. ... Math is a way of solving problems by using numbers and equations.

Ordinary differential equation examples. by GE Sjoden Cited by 6 - An Ordinary Differential Equation (ODE) is an equation that defines a relationship between an independent variable x and a dependent variable y ... Inspection Method Variable Separable Method Homogenous Differential Equations Linear Differential Equation. ... To determine what ...

Step. Calculator Ordinary Differential Equations (ODE) and Systems of ODEs. Calculator applies methods to Without or with initial conditions (Cauchy problem). Do math question. Doing homework can help you learn and understand the material covered in class. Explain mathematic tasks.

Non homogeneous partial differential equations examples - with a corresponding nonhomogeneous partial differential equation, For example, we might take our ... Nonhomogeneous PDE Problems. The methods for finding the Particular Integrals are the same as those for homogeneous linear equations. If f (D,D') is not homogeneous, then