Solve (D^2+5D+6)y=e^x*cos2x

$$( D ^ { 2 } + 5 D + 6 ) y = e ^ { x } \cdot \cos 2 x$$

Solve for D (complex solution)

$$\left\{\begin{matrix}D=\frac{\sqrt{y\left(y+2e^{\left(1-2i\right)x}+2e^{\left(1+2i\right)x}\right)}-5y}{2y}\text{; }D=-\frac{\sqrt{y\left(y+2e^{\left(1-2i\right)x}+2e^{\left(1+2i\right)x}\right)}+5y}{2y}\text{, }&y\neq 0\\D\in \mathrm{C}\text{, }&\frac{-e^{\left(1-2i\right)x}-e^{\left(1+2i\right)x}}{2}=0\text{ and }y=0\end{matrix}\right.$$

Solve for D

$$\left\{\begin{matrix}D=\frac{\sqrt{y\left(4\cos(2x)e^{x}+y\right)}-5y}{2y}\text{; }D=-\frac{\sqrt{y\left(4\cos(2x)e^{x}+y\right)}+5y}{2y}\text{, }&\left(\nexists n_{1}\in \mathrm{Z}\text{ : }x=\frac{\pi \left(2n_{1}+1\right)}{4}\text{ and }y=-4\cos(2x)e^{x}\right)\text{ or }\left(y\leq -4\cos(2x)e^{x}\text{ and }y<0\right)\text{ or }\left(y\geq -4\cos(2x)e^{x}\text{ and }y>0\right)\\D\in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }x=\frac{\pi n_{1}}{2}+\frac{\pi }{4}\text{ and }y=0\end{matrix}\right.$$