17calculus 17calculus
First Order Second Order Laplace Transforms Additional Topics Applications, Practice
Separation of Variables
Linear
Integrating Factors (Linear)
Substitution
Exact Equations
Integrating Factors (Exact)
Linear
Constant Coefficients
Substitution
Reduction of Order
Undetermined Coefficients
Variation of Parameters
Polynomial Coefficients
Cauchy-Euler Equations
Chebyshev Equations
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Classify Differential Equations
Fourier Series
Slope Fields
Wronskian
Existence and Uniqueness
Boundary Value Problems
Euler's Method
Inhomogeneous ODE's
Resonance
Partial Differential Equations
Linear Systems
Exponential Growth/Decay
Population Dynamics
Projectile Motion
Chemical Concentration
Fluids (Mixing)
Practice Problems
Practice Exam List
Exam A1
Exam A3
Exam B2

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17calculus > differential equations > substitution

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Substitution Technique for Differential Equations

Alternate Name For Substitution - Change of Variables

Sometimes we can solve a differential equation by using substitution and changing the variables. This changes the equation into one where we can use a technique we learned previously. There are several different kinds of substitutions that can be done. These are listed below with links to the appropriate discussion.

Types of Substitutions

Given in the Problem Statement

Basic \(v=y'\)

Scaling

Bernoulli Equation Substitution

Homogeneous ODEs

Cauchy-Euler Equation Substitution

Given in the Problem Statement

This, of course, is the easiest kind of substitution you can have. You don't have to analyze or classify the differential equation and then make a determination of what type of substitution will help transform the equation into one that you can solve. The thing to remember here is to make sure and take the derivative of the substitution and substitute for the differentials based on what you are given. For example, if your original variable is x and the new one is t, you can't just say that \(dx = dt\). You need to calculate \(dt\) based on your equation. ( You probably remember this from integration by substitution. The same rule applies here. )

Here is a video showing this technique with an example and lots of detail. This is great place to start to understand what is involved with the technique of substitution.

Dr Chris Tisdell - substitution technique

Okay, after watching that video, you should have a clue what substitution is about and how to accomplish it. The trick now is to learn how to choose a substitution on your own, based on what the differential equation looks like. The next video explains the next three techniques, scaling, Bernoullis equation and homogeneous ODEs. But before viewing that video, let's give a quick overview of these three techniques.

Basic \(v=y'\)

In the special case when you have a second order equation with the dependent variable missing, you can use the substitution \(v=y'\) to reduce it to a first order equation, find \(v\) and then integrate to get \(y\). This technique can be used to solve these two practice problems.

\(y''+y'=t\)

\(ty''-2y'=12t^2\)

Scaling

Sometimes we may just want to scale the equation to make it easier to work with or make it dimensionless. In order to do that, we use these substitutions.

\(x_1 = x/a\)

\(y_1 = y/b\)

where x and y are the old variables

\(x_1\) and \(y_1\) are the new variables, and

a and b are constants

Bernoulli Equation

For equations of the type \(y' = p(x)y + q(x)y^n\), called a Bernoulli equation, we can use the direct substitution \(v = y^{1-n}\), which will turn the equation into a linear equation.
[ Note: This technique uses integrating factors in order to solve the resulting linear equation. ]

Homogeneous ODEs

These first order, linear differential equations can be written in the form, \(y' = f(y/x)\), which should make it obvious that the substitution we use is \(z=y/x\). This is the most common form of substitution taught in first year differential equations.

Now that you have an idea of these three types of substitution, watch this in-depth video to get some examples and detailed explanation on all three topics. If you have not had integrating factors yet, don't worry. You don't need it to understand this video.

MIT OCW - substitution

Cauchy-Euler Equation

When you have an equation of the form \(\displaystyle{ x^2\frac{d^2y}{dx^2} + ax\frac{dy}{dx} + by = 0, ~~~ x > 0 }\), where a and b are constants, you have an Cauchy-Euler equation. One substitution that works here is to let \(t = \ln(x)\). This substitution changes the differential equation into a second order equation with constant coefficients. We discuss this in more detail on a separate page.

As you can tell from the discussion above, there are many types of substitution problems, each with its own technique. We have touched on only a few here. Go through your textbook or search the internet and see if you can find others. If you need a good differential equations textbook, you can find several options at the 17calculus bookstore.

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Practice Problems

Instructions - - Unless otherwise instructed, solve these differential equations using substitution. Give your answers in exact form.

Level A - Basic

Practice A01

\(\displaystyle{\frac{dy}{dx}=\frac{x^2+y^2}{2x^2}}\)

solution

Practice A02

\(\displaystyle{x\frac{dy}{dx}+y=xy^2}\); \(y=1/v\)

solution

Practice A03

\(\displaystyle{y'=x+y}\); \(u=x+y\)

solution

Practice A04

\(\displaystyle{\frac{dy}{dx}=\frac{x+y}{x}}\)

solution

Practice A05

\(y''+y'=t\)

answer

solution

Practice A06

\(ty''-2y'=12t^2\); \(t > 0\)

answer

solution


Level B - Intermediate

Practice B01

\(\displaystyle{\frac{dy}{dx}=\frac{y^2-x^2}{xy}}\)

solution

Practice B02

\(\displaystyle{\frac{dy}{dx}=\frac{y^3+y^2x+yx^2}{yx^2}}\)

solution

Practice B03

\(\displaystyle{\frac{dy}{dx}=\frac{y^4+yx^3}{x^4}}\)

solution

Practice B04

\(\displaystyle{\frac{dy}{dx}=\frac{x^2+3y^2}{2xy}}\)

solution


Level C - Advanced

Practice C01

\(\displaystyle{\frac{dy}{dx}=\frac{y^3+2y^2x-yx^2}{yx^2+x^3}}\)

solution

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