Particular solutions using undetermined coefficients

[Daniel_Feldman]

a)6y''+5y'+3y=(cos(x))^2

b)y''+25y=-4xsin(5x)


I need to know the "guess" for the particular solution. For the first one, I'm not sure because for trig functions, it's usually the sum of that function and it's derivative. So if the right side is just sin(x), the guess is Asin(x)+Bcos(x)....however the guess is different for a polynomial function. Here cos is squared...and it's derivatives oscillate from Acos(x)sin(x) to A (sin^2(x)-cos^2(x))...so I am not quite sure what to use as a guess.

For the second one, I figured that the correct guess for the "x" part is Ax+B, and then for the sin(5x) it would be Csin(5x)+Dcos(5x), so the guess should be (Ax+B)(Csin(5x)+Dcos(5x)), but the computer says that this is wrong. Any ideas?

 

[galactus]

For the first one, Dan, try using the identity \(\displaystyle cos^{2}(x)=\frac{1}{2}(1+cos(2x))\)

 

[Daniel_Feldman]

Awesome, thanks.