- #1

accountkiller

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## Homework Statement

A nonhomogeneous differential equation, a complimentary solution y

_{c}, and a particular solution y

_{p}are given. Find a solution satisfying the initial conditions.

y'' + y = 3x, y(0) = 2, y'(0) = -2, y

_{c}= c

_{1}cos(x) + c

_{2}sin(x), y

_{p}= 3x.

## Homework Equations

y = y

_{p}+ c

_{1}y

_{1}+ ... + c

_{n}y

_{n}

## The Attempt at a Solution

So I first tried solving for the associated homogenous equation y'' + y = 0.

Guessing that y = e

^{rx}, y' = re

^{rx}and y'' = r

^{2}e

^{rx}, so

y'' + y = r

^{2}e

^{rx}+ e

^{rx}= e

^{rx}(r

^{2}+1) = 0, so

r

^{2}+ 1 = 0...

but that gives me a square root of a negative number for r?

For complex numbers, all I have in my notes from lecture is this example:

y

_{1}= e

^{r1x}and y

_{2}= e

^{r2x}with r

_{1}= A + Bi. Solve the DE.

y'' - 2Ay' + (A

^{2}+ B

^{2})y = 0.

I'm not sure how to use that.