ODE - having trouble using method of undetermined coefficients

[amanda_ou812]

Homework Statement

Find a particular solution.
1. y'' +4y = 4 cos (2t) (for this problem, the instructions tell me that the forcing term is a solution of the associated homogeneous solution)
2. y'' + 16 y = 3 sin (4t)

Homework Equations

The Attempt at a Solution

1. I guess that Acos (2t) + Bsin (2t) was a solution but it did not work out. I am thinking that since the roots to the characteristic polynomial are 2i with a multiplicity of two this means that a basis for the homogeneous solution space is { cos (2t) + sin (2t) , t cos (2t) + t sin (2t)}. So I am thinking a better guess should be A t^2 cos (2t) + B t^2 sin (2t). Is this correct?
2. For this one I guessed Acos (4t) + Bsin (4t) was a solution but it did not work out (provided that I did my computation correctly). Any suggestions?


Thanks!

 

[flyingpig]

1) The characteristic equation is

$$r^2 + 4 = 0$$

2i is not a multiplicity of 2...

 

[icystrike]

since your ODE does not have y' term, what would be a better guess?

hint:
y=Acos(4t)
y''=-Acost(4t)

 

[amanda_ou812]

If I choose y =A cos (2t) Then wouldn't y'' = -4 A cos (2t) and then y'' + 4 y = 0

 

[amanda_ou812]

I figured it out