4.414   ODE No. 1414

$$\boxed{{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)=-\frac{(-a^2{(\sinh \left(x\right))}^2-n(n-1))y\left(x\right)}{{(\sinh \left(x\right))}^2}=0}}$$

1. Problem in Latex
2. Mathematica input
3. Maple input

Mathematica: cpu = 1.132144 (sec), leaf count = 231 $$\left\{\{y(x)\to \frac{c_2(-1{)}^{\frac{1}{2}(-2n-1)+1}{\tanh }^2(x{)}^{\frac{1}{4}(-2n-1)+1}{({\tanh }^2(x)-1)}^{\frac{1}{2}(\frac{a+n}{2}+\frac{1}{2}(a+n+1)+\frac{1}{2}(-2n-1)+1)-\frac{1}{2}}{}_2{F}_1(\frac{1}{2}(-2n-1)+\frac{a+n}{2}+1,\frac{1}{2}(-2n-1)+\frac{1}{2}(a+n+1)+1;\frac{1}{2}(-2n-1)+2;{\tanh }^2(x))}{\sqrt{\tanh (x)}}+\frac{c_1{\tanh }^2(x{)}^{\frac{1}{4}(2n+1)}{({\tanh }^2(x)-1)}^{\frac{1}{2}(\frac{a+n}{2}+\frac{1}{2}(a+n+1)+\frac{1}{2}(-2n-1)+1)-\frac{1}{2}}{}_2{F}_1(\frac{a+n}{2},\frac{1}{2}(a+n+1);\frac{1}{2}(2n+1);{\tanh }^2(x))}{\sqrt{\tanh (x)}}\}\right\}$$

Maple: cpu = 0.187 (sec), leaf count = 97 $$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{(\sinh \left(x\right))}^n{}_2\text{F}{}_1(-\frac{a}{2}+\frac{n}{2},\frac{a}{2}+\frac{n}{2};\frac{1}{2};\frac{\cosh \left(2x\right)}{2}+\frac{1}{2})+\mathit{\_}\mathit{C}\mathit{2}{(\sinh \left(x\right))}^n{(2\cosh \left(2x\right)+2)}^{\frac{3}{4}}{}_2\text{F}{}_1(\frac{1}{2}-\frac{a}{2}+\frac{n}{2},\frac{1}{2}+\frac{a}{2}+\frac{n}{2};\frac{3}{2};\frac{\cosh \left(2x\right)}{2}+\frac{1}{2})\sqrt[4]{2\cosh \left(2x\right)-2}\frac{1}{\sqrt{\sinh \left(2x\right)}}\}$$