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17Calculus - Substitution Technique for Solving Differential Equations

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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. Let's get started with the easiest where the substitution is given in the problem statement.

Alternate Name For Substitution - Change of Variables

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 [8mins-14secs]

video by Dr Chris Tisdell

Okay, after watching that video, you should have a clue what substitution is about and how to accomplish it. Although this technique is not complicated, let's work some practice problems, before we go on.

Practice

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

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

Problem Statement

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

Solution

Newcastle University Math - 580 video solution

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\( y' = x + y \); \( u = x + y \)

Problem Statement

\( y' = x + y \); \( u = x + y \)

Solution

Krista King Math - 643 video solution

video by Krista King Math

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The trick now is to learn how to choose a substitution on your own, based on what the differential equation looks like.

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 practice problems.

Practice

Unless otherwise instructed, solve these differential equations using the substitution \(v=y'\). Give your answers in exact form.

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

Problem Statement

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

Final Answer

\( y = t^2/2 - t - c_1e^{-t} + c_2 \)

Problem Statement

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

Solution

Let \(w=y'\) and so \(w'=y''\).
\(y''+y'=t \to w'+w=t\)
This equation is now first-order, linear. So we can use an integrating factor.
\(\mu=exp \int{1~dt} = e^t \)
\(\displaystyle{ \frac{d}{dt}[we^t] = te^t }\)
\(we^t = \int{ te^t~dt}\)
Now we use integration by parts on the integral.
\(u=t \to du=dt\) and \(dv=e^tdt \to v=e^t\)
\(\begin{array}{rcl} we^t &=& te^t-\int{e^t~dt} \\ &=& te^t-e^t+c_1 \\ w &=& t-1+c_1e^{-t} \\ y' &=& t-1+c_1e^{-t} \\ y &=& \int{t-1+c_1e^{-t}~dt} \\ y &=& t^2/2-t-c_1e^{-t}+c_2 \end{array}\)

Final Answer

\( y = t^2/2 - t - c_1e^{-t} + c_2 \)

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

Problem Statement

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

Final Answer

\( y = 4t^3\ln t + kt^3 + c \)

Problem Statement

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

Solution

Let \(w=y'\), so \(w'=y''\).
\(ty''-2y'=12t^2 \to tw'-2w=12t^2 \to \) \(w'-(2/t)w=12t\)
This last equation is first-order, linear, so we can use an integrating factor.
\(\mu=exp \int{-2/t~dt} = e^{-2\ln t}=t^{-2}\)
\(\displaystyle{ \frac{d}{dt}[t^{-2}w] = 12/t }\)

\( t^{-2}w = 12\ln t + c_1 \)

\( w = 12t^{2}\ln t+c_1t^2 \)

\( y = \int{ 12t^2 \ln t+c_1t^2~dt} \)

\(\displaystyle{ y = 12\left[ \frac{t^3}{3}\ln t - \int{t^2 / 3 ~dt} \right] + c_1(t^3)/3 }\)

\( y = 4t^3\ln t - 4t^3/3 +c_1t^3/3+c_2 \)

\( y = 4t^3\ln t + kt^3+c \)

Notes:
1. We used integration by parts on the integral \(\int{t^2\ln t~dt}\) letting
\(u=\ln t \to du=(1/t)dt\) and \(dv=t^2~dt \to v=t^3/3\).
2. In the last step, we combined the \(t^3\) terms and replaced the constant. This was not necessary but it made the final answer more compact and easier to read. Doing this step would also allow us to more easily determine the particular solution, if initial conditions had been given.

Final Answer

\( y = 4t^3\ln t + kt^3 + c \)

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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

Let's watch a video clip, discussing this in detail.

MIT OCW - introduction to substitution and scaling [14mins-18secs]

video by MIT OCW

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. Let's watch a video clip discussing this.

MIT OCW - Homogenous ODEs [24mins-53secs]

video by MIT OCW

Now try solving these homogeneous ODEs.

Practice

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

Problem Statement

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

Solution

Newcastle University Math - 579 video solution

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

Problem Statement

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

Solution

PatrickJMT - 575 video solution

video by PatrickJMT

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

Problem Statement

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

Solution

PatrickJMT - 576 video solution

video by PatrickJMT

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

Problem Statement

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

Solution

PatrickJMT - 577 video solution

video by PatrickJMT

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

Problem Statement

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

Solution

PatrickJMT - 578 video solution

video by PatrickJMT

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

Problem Statement

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

Solution

Khan Academy - 644 video solution

video by Khan Academy

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

Problem Statement

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

Solution

Khan Academy - 645 video solution

video by Khan Academy

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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.

Really UNDERSTAND Differential Equations

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Given in the Problem Statement - Practice

Basic - Practice

Scaling

Homogeneous ODEs - Practice

Cauchy-Euler Equation

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