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hw28

The document outlines homework assignments for a Linear Algebra course, focusing on solving various differential equations. It includes specific problems requiring general solutions and methods for solving inhomogeneous equations using operator inversion and a cookbook approach. The document emphasizes the application of differential equations in time-dependent scenarios and provides examples of solution techniques.

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2 views2 pages

hw28

The document outlines homework assignments for a Linear Algebra course, focusing on solving various differential equations. It includes specific problems requiring general solutions and methods for solving inhomogeneous equations using operator inversion and a cookbook approach. The document emphasizes the application of differential equations in time-dependent scenarios and provides examples of solution techniques.

Uploaded by

moien
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Math 21b: Linear Algebra Spring 2016

Homework 28: Differential equations


This homework is due on Monday, April 18, respectively on Tuesday, April 19, 2016.

1 a) Find the general solution to f 0 = sin(17t) + 4et.


b) Find the general solution to f 00 = 7 cos(20t).
2 a) Find the solution to f 0 − 3f = e5t by inverting the operator
D − 3.
b) Find the solution again, this time with the cookbook: find
first the solution of f 0 − 3f = 0, then make a guess for a special
solution.
3 a) Find the general solution of f 00 + 9f = e2t.
b) Find the general solution of f 00 + 9f = cos(2t).
c) Find the general solution of f 00 + 9f = cos(3t).
d) Why does the solution in c) behave differently than the solution
in b).
4 a) Find the general solution to f 00 −10f 0 +25f = cos(3t)+sin(4t).
b) Find the general solution to f 00 − 10f 0 + 25f = cos(5t)
5 a) Find the general solution to f 000 − 2f 00 − f 0 + 2f = sin(3t).
b) Find the general solution to f 000 − 2f 00 − f 0 + 2f = et + e−t.
Differential equations

To solve a differential equation (D − 3)f = 0 we write it as


f 0 = 3f and know that f = Ce3f is a solution, where C is a
constant. In this section, we use the variable t and write f (t)
as f (t) often represents a quantity which changes in time. We
write f 0(t) = d/dtf (t) or Df to simplify writing. To solve a
differential equation like (D2 + 2D − 15)f = 0, we factor the
polynomial and write (D − 3)(D + 5)f = 0 which has solutions
satisfying (D + 5)f = 0 and (D − 3)f = 0. The linear space of
solutions to (D − 3)(D + 5)f = 0 is therefore given by functions
f (t) = Ae−5t + Be3t, where A, B are constants. To find the
solution to the inhomogeneous equation like (D − 3)f = et, we
can invert the operator D − λ = D − 3. Its inverse is explicitly
given by Z t
f (t) = Ceλt + eλt 0 e−λsg(s) ds .
In the case g(t) = et, this gives f (t) = Ce3t + 2et. To solve
(D−3)(D+5)f = et, we have now to solve (D+5)f = Ce3t +2et
and therefore invert D + 5 with the above formula. We get
the general solution f (t) = Ae3t + Be−5t + 2et. This general
operator method works universally, but it can be a bit tedious.
In a handout, we show how to get a special solution using a
“cookbook recipe”. This is how engineers solve the systems.

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