Question

Use the principle of superposition to find a particular solution of y"+y'+y=xe^(x)+e^(-x)(1+2x).

Answer 1

@SithsAndGiggles

Answer 2

For this one, none of the characteristic solutions will conflict with the particular solutions, since
\[\begin{align*}y''+y'+y=0~~\implies~~y_c&=C_1e^{-(1-\sqrt3)/2x}+C_2e^{-(1+\sqrt3)/2x}\\[1ex]
&=C_1e^{-x}\cos\sqrt3x+C_2e^{-x}\sin\sqrt3x\end{align*}\]
For the particular solution, you can try \(y_{p_1}=(a_0+a_1x)e^x\) and \(y_{p_2}=(b_0+b_1x)e^{-x}\).

Answer 3

Thank you!