The method of undetermined coeficients for a non-homogenous linear ODE with a rational RHS Consider the equation $$\ddot x-5\dot x+4x=\frac{9e^t}{1+e^{-3t}}$$ The characteristic roots are one and four. The solution to the homogenous equation is $C_1e^t+C_2e^{4t}$. How can we find a particular solution to the equation? Can we use the method of undetermined coefficients? If so what would be the form of the solution?

Set $$x(t) = C_1(t) x_1(t),$$ where $x_i(t) = e^{s_i t}$, $s_i = \\{1,4\\}$ (you can chose any of the roots). Differentiate now to obtain $x'$ and $x''$ and substitute back into the original ODE. Take into account that $x_1'' - 5x_1'+ 4x_1 = 0$ and solve for $C_1$.

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