Rozwiąż y^2-4y^prime+y=sin^2x

$$y ^ { 2 } - 4 y ^ { \prime } + y = \sin ^ { 2 } x$$

Rozwiąż względem x (complex solution)

$$x=-\frac{i\ln(2\sqrt{y\left(y^{3}+2y^{2}-1\right)}-2y^{2}-2y+1)}{2}+\pi n_{1}\text{, }n_{1}\in \mathrm{Z}$$

$$x=-\frac{i\ln(-2\sqrt{y\left(y^{3}+2y^{2}-1\right)}-2y^{2}-2y+1)}{2}+\pi n_{2}\text{, }n_{2}\in \mathrm{Z}$$

Rozwiąż względem y (complex solution)

$$y=\frac{\sqrt{-2\cos(2x)+3}-1}{2}$$

$$y=\frac{-\sqrt{-2\cos(2x)+3}-1}{2}$$

Rozwiąż względem x

$$x=\arcsin(\sqrt{y\left(y+1\right)})+\pi n_{1}\text{, }n_{1}\in \mathrm{Z}$$

$$x=-\arcsin(\sqrt{y\left(y+1\right)})+\pi n_{2}\text{, }n_{2}\in \mathrm{Z}\text{, }\left(y\geq \frac{-\sqrt{5}-1}{2}\text{ and }y\leq -1\right)\text{ or }\left(y\geq 0\text{ and }y\leq \frac{\sqrt{5}-1}{2}\right)$$

Rozwiąż względem y

$$y=\frac{\sqrt{4\left(\sin(x)\right)^{2}+1}-1}{2}$$

$$y=\frac{-\sqrt{4\left(\sin(x)\right)^{2}+1}-1}{2}$$