ແກ້ d^2y/dx^2-cotxdy/dx-(1-cotx)y=e^xsinx

$$\frac { d ^ { 2 } y } { d x ^ { 2 } } - \cot x \frac { d y } { d x } - ( 1 - \cot x ) y = e ^ { x } \sin x$$

ແກ້ສຳລັບ y (complex solution)

$$\left\{\begin{matrix}y=-\frac{ie^{\left(1-2i\right)x}+ie^{\left(1+2i\right)x}-2ie^{x}}{2\left(\left(-1+i\right)e^{ix}+\left(1+i\right)e^{-ix}\right)}\text{, }&\nexists n_{3}\in \mathrm{Z}\text{ : }\left(x=\pi n_{3}+\frac{\pi }{4}\text{ or }x=\pi n_{3}\right)\\y\in \mathrm{C}\text{, }&ie^{\left(1-2i\right)x}+ie^{\left(1+2i\right)x}-2ie^{x}=0\text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }x=\pi n_{2}+\frac{\pi }{4}\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\end{matrix}\right,$$

ແກ້ສຳລັບ y

$$y=\frac{\left(\sin(x)\right)^{2}e^{x}}{\cos(x)-\sin(x)}$$

$$\nexists n_{2}\in \mathrm{Z}\text{ : }\left(x=\pi n_{2}+\frac{\pi }{4}\text{ or }x=\pi n_{2}\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}$$