Solve (D^2-4D+3)y=2xe^3x+3e^xcos2x

$$( D ^ { 2 } - 4 D + 3 ) y = 2 x e ^ { 3 x } + 3 e ^ { x } \cos 2 x$$

Solve for D (complex solution)

$$\left\{\begin{matrix}D=\frac{\sqrt{2y\left(4xe^{3x}+2y+3e^{\left(1-2i\right)x}+3e^{\left(1+2i\right)x}\right)}+4y}{2y}\text{; }D=\frac{-\sqrt{2y\left(4xe^{3x}+2y+3e^{\left(1-2i\right)x}+3e^{\left(1+2i\right)x}\right)}+4y}{2y}\text{, }&y\neq 0\\D\in \mathrm{C}\text{, }&-2xe^{3x}-\frac{3e^{\left(1-2i\right)x}}{2}-\frac{3e^{\left(1+2i\right)x}}{2}=0\text{ and }y=0\end{matrix}\right.$$

Solve for D

$$\left\{\begin{matrix}D=\frac{\sqrt{y\left(3\cos(2x)e^{x}+2xe^{3x}+y\right)}+2y}{y}\text{; }D=\frac{-\sqrt{y\left(3\cos(2x)e^{x}+2xe^{3x}+y\right)}+2y}{y}\text{, }&\left(3\cos(2x)e^{x}+2xe^{3x}\neq 0\text{ and }y=-3\cos(2x)e^{x}-2xe^{3x}\right)\text{ or }\left(y\leq -3\cos(2x)e^{x}-2xe^{3x}\text{ and }y<0\right)\text{ or }\left(y\geq -3\cos(2x)e^{x}-2xe^{3x}\text{ and }y>0\right)\\D\in \mathrm{R}\text{, }&-3\cos(2x)e^{x}-2xe^{3x}=0\text{ and }y=0\end{matrix}\right.$$