2.1025   ODE No. 1025

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

$$y^{″}(x)-y(x)(a+(m-1)m{\sec }^2(x)+(n-1)n{\csc }^2(x))=0$$ ✓ Mathematica : cpu = 1.20371 (sec), leaf count = 158

$$\left\{\{y(x)\to \frac{(-1{)}^{-m}{\cos }^2(x{)}^{-\frac{m}{2}-\frac{1}{4}}{(-{\sin }^2(x))}^{n/2}(c_1(-1{)}^m{\cos }^2(x{)}^{m+\frac{1}{2}}{}_2{F}_1(\frac{1}{2}(m+n-\sqrt{-a}),\frac{1}{2}(m+n+\sqrt{-a});m+\frac{1}{2};{\cos }^2(x))+i{c}_2{\cos }^2(x){}_2{F}_1(\frac{1}{2}(-m+n-\sqrt{-a}+1),\frac{1}{2}(-m+n+\sqrt{-a}+1);\frac{3}{2}-m;{\cos }^2(x)))}{\sqrt{\cos (x)}}\}\right\}$$

✓ Maple : cpu = 0.243 (sec), leaf count = 102

$$\{y\left(x\right)={(\sin \left(x\right))}^n({(\cos \left(x\right))}^{-m+1}{}_2\text{F}{}_1(\frac{n}{2}-\frac{m}{2}+\frac{i}{2}\sqrt{a}+\frac{1}{2},\frac{n}{2}-\frac{m}{2}-\frac{i}{2}\sqrt{a}+\frac{1}{2};\frac{3}{2}-m;{(\cos \left(x\right))}^2)\mathit{\_}\mathit{C}\mathit{2}+{(\cos \left(x\right))}^m{}_2\text{F}{}_1(\frac{n}{2}+\frac{m}{2}+\frac{i}{2}\sqrt{a},\frac{n}{2}+\frac{m}{2}-\frac{i}{2}\sqrt{a};\frac{1}{2}+m;{(\cos \left(x\right))}^2)\mathit{\_}\mathit{C}\mathit{1})\}$$