Solve psi^2=2/10sin^2(2pix/10)

$$\psi ^ { 2 } = \frac { 2 } { 10 } \sin ^ { 2 } ( 2 \pi x / 10 )$$

Solve for x (complex solution)

$$x=10n_{2}+\left(-5i\right)\pi ^{-1}\ln(i\times 5^{\frac{1}{2}}\psi +i\left(5\psi ^{2}-1\right)^{\frac{1}{2}})\text{, }n_{2}\in \mathrm{Z}$$

$$x=10n_{1}+\left(-5i\right)\pi ^{-1}\ln(i\times 5^{\frac{1}{2}}\psi +\left(-i\right)\left(5\psi ^{2}-1\right)^{\frac{1}{2}})\text{, }n_{1}\in \mathrm{Z}$$

$$x=10n_{14}+\left(-5i\right)\pi ^{-1}\ln(\left(-i\right)\times 5^{\frac{1}{2}}\psi +i\left(5\psi ^{2}-1\right)^{\frac{1}{2}})\text{, }n_{14}\in \mathrm{Z}$$

$$x=10n_{13}+\left(-5i\right)\pi ^{-1}\ln(\left(-i\right)\times 5^{\frac{1}{2}}\psi +\left(-i\right)\left(5\psi ^{2}-1\right)^{\frac{1}{2}})\text{, }n_{13}\in \mathrm{Z}$$

Solve for ψ (complex solution)

$$\psi =-\frac{\sqrt{10\left(-\cos(\frac{2\pi x}{5})+1\right)}}{10}$$

$$\psi =\frac{\sqrt{10\left(-\cos(\frac{2\pi x}{5})+1\right)}}{10}$$

Solve for ψ

$$\psi =\frac{\sqrt{5}\sin(\frac{\pi x}{5})}{5}$$

$$\psi =-\frac{\sqrt{5}\sin(\frac{\pi x}{5})}{5}$$