Solve (D^2-9)y=e^3xcosx

$$( D ^ { 2 } - 9 ) y = e ^ { 3 x } \cos x$$

Solve for D (complex solution)

$$\left\{\begin{matrix}D=-\frac{iy^{-\frac{1}{2}}\sqrt{-36y-2e^{\left(3-i\right)x}-2e^{\left(3+i\right)x}}}{2}\text{; }D=\frac{iy^{-\frac{1}{2}}\sqrt{-36y-2e^{\left(3-i\right)x}-2e^{\left(3+i\right)x}}}{2}\text{, }&y\neq 0\\D\in \mathrm{C}\text{, }&\frac{-e^{\left(3-i\right)x}-e^{\left(3+i\right)x}}{2}=0\text{ and }y=0\end{matrix}\right.$$

Solve for D

$$\left\{\begin{matrix}D=\sqrt{\frac{\cos(x)e^{3x}}{y}+9}\text{; }D=-\sqrt{\frac{\cos(x)e^{3x}}{y}+9}\text{, }&\left(y<0\text{ and }\exists n_{3}\in \mathrm{Z}\text{ : }\left(x\geq \frac{\pi \left(4n_{3}+1\right)}{2}\text{ and }x\leq \frac{\pi \left(4n_{3}+3\right)}{2}\right)\right)\text{ or }\left(y\leq -\frac{\cos(x)\left(e^{x}\right)^{3}}{9}\text{ and }\exists n_{3}\in \mathrm{Z}\text{ : }\left(x>\frac{\pi \left(4n_{3}+3\right)}{2}\text{ and }x<\frac{\pi \left(4n_{3}+5\right)}{2}\right)\right)\text{ or }\left(y>0\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\left(x\geq \frac{\pi \left(4n_{4}+3\right)}{2}\text{ and }x\leq \frac{\pi \left(4n_{4}+5\right)}{2}\right)\right)\text{ or }\left(y=-\frac{\cos(x)\left(e^{x}\right)^{3}}{9}\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }x=\frac{\pi \left(2n_{1}+1\right)}{2}\right)\text{ or }\left(y\geq -\frac{\cos(x)\left(e^{x}\right)^{3}}{9}\text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }\left(x>\frac{\pi \left(4n_{2}+1\right)}{2}\text{ and }x<\frac{\pi \left(4n_{2}+3\right)}{2}\right)\right)\text{ or }\left(y\neq 0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }x=\frac{\pi \left(2n_{1}+1\right)}{2}\right)\\D\in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{\pi }{2}\text{ and }y=0\end{matrix}\right.$$