Question on initial value problem

[cue928]

I am working on the following initial value problem:

y'' + 4y = 4, y(0) = 1, y'(0) = 1

The method they show is:
(1): y = A cos x + B sin x
A = 1, (2): y = 1 + A cos x + B sin x
y(0) = 1 + A = 0
y'(0) = B = 1
Final answer is y(x) = 1 + sin(x)

The problem I don't see is the first equation (1). I had y = A cos(2x) + B sin(2x) by solving the LHS. The, I am confused on how they proceed with writing the equation (2). I've done IVP's before without problems, so I am confused what I am missing here.

 

[summerwind]

I'm happy to help out if you correct a few errors first.

In (1): you have written 'A = 1.' But in fact your final answer has A = 0. Are you using 'A' to mean different things?

In (2): we know that y(0) = 1, so it can't be the case that 1 + A = 0.

Finally, your answer y(x) = 1 + sin (x) does not satisfy the equation y'' + 4y = 4.

 

[tiny-tim]

hi cue928!

y'' + 4y = 4, y(0) = 1, y'(0) = 1
(2): y = 1 + A cos x + B sin x

yes you're right …

clearly, (2) should be y = 1 + A cos 2x + B sin 2x