# Second Order Differential Equations

## Homework Statement

Consider the second order differential equation y'' - 4y' + 4y = f(x)
Find a particular solution if f(x) = 25cos(x)

## Homework Equations

I believe for this type of question I should let y = Asin(x) + B cos(x)
Hence y' = Acos(x) - Bsin(x) and
y'' = -Asin(x) - Bsin(x)

## The Attempt at a Solution

I think everything above is right, and for the rest of the question I should just be able to substitute back into the original formula and find values for A and B.
The problem is when I do substitute back in all I get is a mess and a headache!

Here is what I got by substitution:
-Asin(x) - Bcos(x) - 4[Acos(x) - Bsin(x)] + 4[Asin(x) + Bcos(x)] = 25cos(x)
Hence
-Asin(x) + 4[Asin(x) + Bsin(x)] - Bcos(x) - 4[Acos(x) - Bcos(x)] = 25cos(x)

That's about all I think I can do and I have no idea how to obtain values for A and B. I know the answer is y(p) = 3cos(x) - 4sin(x).
How to get this answer from the above I have no idea, I'm obviously missing something.
Thanks in advance for any advice or help.

## Answers and Replies

Integral
Staff Emeritus
Science Advisor
Gold Member
You are just about there. Gather your sinx terms then factor the sin, same for the cosx terms, now, since sin x does not appear in the particular solution set its coefficient to zero, solve for A in terms of B. Set the coefficient for the cos x term = 25 and find B.