Lesson 7, Substitution Methods

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Handout Lesson 7, Substitution Methods

Textbook Section(s).

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This lesson is based on Section 1.6 of your textbook by Edwards, Penney, and Calvis.

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Overview of Substitution Methods for Solving Differential Equations.

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When you studied \(u\)-substitution in calculus, here are some big-picture lessons you should have gleaned:

Practice is necessary for building intuition when trying to evaluate an integral using substitution. (Which integrals can be evaluated using substitution? Which substitution should you try?)

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Not every integral can be evaluated using substitution.

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There are some standard categories of integrals that can be evaluated using substitution and if we recognize that an integral is in one of these categories, then the required substitution is straight-forward.

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The same principles will apply today as we discuss substitution methods for solving some differential equations.

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Recall: If

\begin{equation*} y=\beta(v,x) \end{equation*}

is a function of two variables, then the chain rule from multi-variable calculus gives us that

\begin{equation*} \frac{dy}{dx}=\beta_x\frac{dx}{dx}+\beta_v\frac{dv}{dx} \end{equation*}

which implies

\begin{equation*} \frac{dy}{dx}=\beta_x+\beta_v\frac{dv}{dx} \end{equation*}

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Let’s take a look at an example that uses substitution. This example was taken from T. Bazett’s website in the discussion for section 1.5.1.

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Example 41. Using substituion to solve a differential equation.

Solve the differential equation.

\begin{equation*} \frac{dy}{dx}=(x-y+1)^2 \end{equation*}

This differential equation is NOT and is NOT . The intuition that we developed when solving integrals by \(u\)-substitution, tells us that it might be a good idea to try the substitution

\begin{equation*} v=x-y+1 \end{equation*}

because this is the inside function.

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Homogeneous Differential Equations.

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Definition 42.

A differential equation of the form

\begin{equation*} \frac{dy}{dx}=f\left(\frac{y}{x} \right) \end{equation*}

is a homogeneous differential equation.

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(We will talk about a different kind of homogeneous differential equation later in the semester. The two types of homogeneous equations are not related and you must distinguish them from the form.)

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For homogeneous differential equations, the substitution \(v=\frac{y}{x}\) might be helpful.

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Example 43. Solving homogeneous differential equations.

(Number 2 from Section 1.6 of your textbook by Edwards, et.al.)

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Solve the differential equation.

\begin{equation*} 2xy\frac{dy}{dx}=x^2+2y^2 \end{equation*}

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When working with homogeneous equations, it is really important that the form \(f\left(\frac{x}{y} \right)\) is evident. When the form is evident, the substitution is more efficient, and you are less likely to make errors.

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Example 44. Solving homogeneous differential equations.

(Number 4 from Section 1.6 of your textbook by Edwards, et.al.)

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Solve the differential equation.

\begin{equation*} (x-y)\frac{dy}{dx}=x+y \end{equation*}

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Bernoulli Differential Equations.

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Definition 45.

A differential equation of the form

\begin{gather} \frac{dy}{dx}+P(x)y=Q(x)y^n\tag{✶} \end{gather}

is a Bernoulli Differential Equation

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If \(n=0\) or \(n=1\) in (✶), then the equation is a first-order linear differential equation and can be solved using techniques learned previously. If \(n \not\in\set{0,1}\text{,}\) then the following strategy may be helpful.