Understanding partial solutions

[azuralshadow]

hello, I've spent a good couple hours diving back into the world of differential equations after being out of the game for a good 2 years. I started getting a hang of solving them till i came across this problem:
Solve the following differential equation with the 3 given cases, all of the systems have a sinusoidal input 'y' and start undeflected and at rest.
X'' + 2(A)(B)X' + (B^2)X = y
Initial conditions x' = 0 , x = 0 , y = sin(t)

Case 1: A = 0.5 , B = 10
Case 2: A = 1.0 , B = 10
Case 3: A = 2.0 , B = 10

Honestly the part I am having the hardest time doing is figuring out how to make a good guess at a particle solution after that i understand how to get to a general solution.

 

[Sudharaka]

hello, I've spent a good couple hours diving back into the world of differential equations after being out of the game for a good 2 years. I started getting a hang of solving them till i came across this problem:
Solve the following differential equation with the 3 given cases, all of the systems have a sinusoidal input 'y' and start undeflected and at rest.
X'' + 2(A)(B)X' + (B^2)X = y
Initial conditions x' = 0 , x = 0 , y = sin(t)

Case 1: A = 0.5 , B = 10
Case 2: A = 1.0 , B = 10
Case 3: A = 2.0 , B = 10

Honestly the part I am having the hardest time doing is figuring out how to make a good guess at a particle solution after that i understand how to get to a general solution.

Hi azuralshadow, :)

This can be solved using the method of Undetermined Coefficients. Take the particular solution as \(y_{p}=C\sin t+D\cos t\) where \(C\) and \(D\) are constants to be determined. You can find some useful ideas here(Refer to example 3).

Kind Regards,
Sudharaka.