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Section 1.8

LINEAR EQUATIONS

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Introduction

First, take a few minutes to read the introduction on p. 112.

1. What is the only technique we have learned so far for actually solving DEs?

2. Why aren’t you going to learn how to solve every DE in this course? (Is Dr. Hall being lazy?)

3. What tools and techniques are available if you don’t know how to solve a DE? Which of them will give you exact solutions?

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Definition

A first-order* differential equation is linear if it can be written in the form

where a(t) and b(t) are any functions of t.Examples:

NOT linear:

*First order means that the equation contains onlyfirst derivatives (dy/dt, not d2y/dt2).

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dydt= a(t)y+ b(t)

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dydt− ty+ t

4 − y1+ t2

= 0

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dydt= e ty+ cos t

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dydt= y(1− y)

Find a(t) and b(t).

Explainwhy thisisn’t linear.

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Match-making!

Read p. 114, top. Choose all that apply:

A. dy/dt = 3y - 7

B. dy/dt = cost + y2

C. dy/dt = yet

D. dy/dt = 0

E. dy/dt = t2(1 + y)

1. linear homogeneous

2. linear nonhomogeneous

3. linear constant-coefficient

4. none of the above

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Answers

A23, B4, C1, D13, E2

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The homogeneous case

• The linearity principle: If yh(t) is a solution of a linear homogenous DE, then so is any constant multiple of yh(t). Explain why this is true.

• What technique can you use to solve a linear homogeneous DE? Explain.

• Solve

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dydt= −t2y

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The nonhomogeneous case

Suppose yh(t) is a solution of the homogeneous DE

and suppose yp(t) is a particular solution of the nonhomogeneous DE

Let y(t) = yh(t)+yp(t). Show that y(t) is a solution of the nonhomogeneous DE above.

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dydt+ t2y = 0

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dydt+ t2y = 4

Hint: plug y(t) into the nonhomogeneous equation.

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YOU make the call!A tale of two equations:

(NDE)

which is a nonhomogeneous DE (assume b(t)≠0), and(AHDE)

which is its associated homogeneous DE.

Make a hypothesis about the relationship between solutions of (NDE) and (AHDE). Check your hypothesis by plugging into (NDE).

Try to make your hypothesis as general as possible.

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dydt= a(t)y+ b(t)

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dydt= a(t)y

Hint: use the linearity principle if you haven’t already.

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Solving Linear Equations

Three-step procedure:

1) Find the general solution of the associated homogeneous DE.

2) Find one particular solution of the nonhomogeneous DE.

3) The general solution of the nonhomogeneous DE is the sum of the solutions you found in steps (1) and (2).

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Arg! Step (2) is tricky

We haven’t developed any general techniques for finding even one solution of a nonhomogeneous DE.

Hmmm…Maybe we could just, like, guess a solution. Maybe

we’d get lucky! After all, we only need one.This extremely precise scientific technique is known as

the METHOD OF LUCKY GUESS.

(To tell the truth, a lot of math and science works this way: you have a hunch that something is true, then you try to prove (or disprove) it.)

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The lucky guessRead p. 118.• Write down the NDE, its AHDE, and the general

solution to the AHDE.• What initial guess do the authors make?• Why is it not right?• What is their next guess?• Explain how they finish the problem.• Guess a particular solution to p. 123 #1. • Did you get lucky?• Finish the problem.

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Another lucky guess

Read p. 119• Why do they guess a sine plus a cosine, rather than

just a sine or a cosine? Try yp(t) = cos 3t if you’re not convinced.

• Explain each step in the computation. In particular, explain where the “simultaneous algebraic equations” come from.

• What guesses should you make for p. 123, #3 and #4?

• Exercise: do p. 123 #3• The method we just developed is called The Method

of Undetermined Coefficients ( and are the undetermined coefficients).

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Qualitative analysis

• Use the HPGSolver to analyze solutions to

qualitatively. What is the long-term behavior of every solution?

• Read p. 120-1 for an explanation of this.• Now tell me about the long-term behavior of solutions

to

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dydt= −2y+ cos3t

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dydt= −5y+ b(t)

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dydt= 5y+ b(t)anda) b)