Second order nonhomogeneous ODE

[yaro99]

Homework Statement

y''+3y'+3.25=3cost-1.5sint

Homework Equations

y_h = e^(a/2)t(Acost+Bsint)
y_p = Kcos(ωt)+Msin(ωt) [when r(x)=kcos(ωt) or ksin(ωt)]

The Attempt at a Solution

I got the homogeneous solution, which is e^-1.5t(Acost+Bsint)
but I am having trouble with the particular solution.

I tried the above equation, making y_p=K₁cos(ωt)+M₁sin(ωt)+K₂cos(ωt)+M₂sin(ωt)
since there are 2 trig functions as r(t).
I couldn't solve for the variables by plugging into the original equation because I was left with 4 variables and only 2 equations.

EDIT: realized I wasn't consistent with my independent variable, made them all t's instead of t's and x's

 

[djh101]

K₁cos(ωx) + K₂cos(ωx) = (K₁ + K₂)cos(ωx). Having two sin/cos functions is redundant, since they are linear combinations (in this case, they are the same function, entirely).

 

[yaro99]

K₁cos(ωx) + K₂cos(ωx) = (K₁ + K₂)cos(ωx). Having two sin/cos functions is redundant, since they are linear combinations (in this case, they are the same function, entirely).

Ah I get it now. I solved for the values (M₁+M₂) and (K₁+K₂) instead of the variables individually and got the right answer. Thanks!