Linear Algebra: Solving a system of equations for damped oscillation

[mahrap]

So we are given two equations:

$$ \ddot{x} - \dot{x} - x = cost (t) $$

and

$$ x(t) = a sin(t) + b cos(t) $$

The question asks to find a and b.

How would one go about doing this? I thought maybe substituting the $$ cos(t) $$ from equation 1 into equation 2 would work but then what system of equations would I have to solve? I am completely clueless on how to set up this problem. Any suggestions and hints are appreciated.

 

[dinospamoni]

Was there any additional information?

Try taking the first and second derivatives of x

 

[mahrap]

There was not much additional information which would have helped me arrive at a solution. What would I do after taking the second derivative of x with respect to t? Plug it into equation 1? But then How would I solve my equations then?

 

[dinospamoni]

Ok, just wondering.

Take the first and second derivatives of x, then plug those into the first equation. You should see from there