1. ## Undetermined Coefficients question

$y^{\prime \prime} + 81 y = 9 \sin(9 x) + 9 + e^{9 x}$

My Yp expression is:
$
C_{1}x\cos(9 x) + C_{2} x\sin(9 x) + C_{3} + C_{4}e^{9 x}
$

But for this one,
$y^{\prime \prime} - 25 y = 5 \cos(5 x) + 5 + e^{5 x}$
its:
$C_{1}\cos(5 x) + C_{2} \sin(5 x) + C_{3} + C_{4}x e^{5 x}$

For the second one, the expression for cos(5x) is written as [tex] $C_{1}\cos(5 x) + C_{2} \sin(5 x)$

but for the first one, sin(9x) its :
$
C_{1}x\cos(9 x) + C_{2} x\sin(9 x)
$

Why does the second instance have an x infront of the cos and sin? Whats the rule here?

2. The rule here is that functions that show up in the homogeneous solution (that is, the differential equation LHS = 0), need to be "enhanced" in order not to be annihilated by the LHS's operator when you plug it into the DE. Notice that in your second yp, there's an x multiplying the exponential. That's because in that case, the exponential is already a part of the homogeneous solution. In the first problem, the trig functions are both in the homogeneous solution. Make sense?

3. Oh Thank you!
However, when I solve my homo equation for the first one, I get
$ACos(18x)+BSin(18x)$
Which isnt quite whats on the RHS. Does it matter whats inside the cos and sin?

4. It does matter what's inside the trig functions, and I don't think your homogeneous solution is correct. Check it again, if you would, please.