undetermined coefficient Solution, distinct roots, higher-order Differential

Question

Solve $y(x)^{(4)}-y(x)=\sin(x)$.

So far my $y_c$ is $C_1e^-x + C_2e^x + C_3\cos x + C_4\sin x$ and my $y_p$ is $Ax\cos x+Bx\sin x$.

Any suggestions would be appreciated.

Answer 1

After plugging in the particular solution you should get

$$ (4A\sin x - 4B\cos x + Ax\cos x + Bx\sin x) - (Ax\cos x + Bx\sin x) = \sin x $$

$$ 4A\sin x - 4B\sin x = \sin x $$

Matching coefficients gives $$ 4A = 0 $$ $$ -4B = 1 $$

Answer 2

Plug your $y_p$ back into your original differential equation and solve for $A$ and $B$. Your general solution is $y_c+y_p$.

Specifically for this problem, solve $(Ax\cos x +Bx\sin x)^{(4)}-(Ax\cos x +Bx\sin x)=\sin x$ for $A$ and $B$. Note that you can substitute values in for $x$ until you get enough equations to solve for $A$ and $B$.