\(\beta^2=\frac{N}{(y_c^2A+I_c+(e_0a)^2N)}\)
for simplicity we solve for n=1 case
\(y=\frac{1}{8}\cdot\left(\sin\left(\pi\cdot\frac{x}{2\cdot5}\right)-\sqrt{\sin^2\left(\pi\cdot\frac{x}{2\cdot5}\right)-\frac{16a^2}{3}}\right)\)