4.25   ODE No. 1025

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

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

Mathematica: cpu = 0.954121 (sec), leaf count = 615 $$\left\{\{y(x)\to \frac{c_2(-1{)}^{\frac{1}{2}(-2m-1)+1}{\cos }^2(x{)}^{\frac{1}{4}(-2m-1)+1}{({\cos }^2(x)-1)}^{\frac{1}{2}(\frac{4am+4\sqrt{-a}{n}^2+4an-4\sqrt{-a}n+4(-a{)}^{3/2}+8\sqrt{-a}a+\sqrt{-a}+4m{n}^2-4mn+m+4n^3-4n^2+n}{8a+8n^2-8n+2}+\frac{1}{2}(-\sqrt{-a}+m+n)+\frac{1}{2}(-2m-1)+1)-\frac{1}{4}}{}_2{F}_1(\frac{1}{2}(-2m-1)+\frac{1}{2}(m+n-\sqrt{-a})+1,\frac{1}{2}(-2m-1)+\frac{4n^3+4m{n}^2+4\sqrt{-a}{n}^2-4n^2+4an-4mn-4\sqrt{-a}n+n+4(-a{)}^{3/2}+8\sqrt{-a}a+4am+m+\sqrt{-a}}{8n^2-8n+8a+2}+1;\frac{1}{2}(-2m-1)+2;{\cos }^2(x))}{\sqrt{\cos (x)}}+\frac{c_1{\cos }^2(x{)}^{\frac{1}{4}(2m+1)}{({\cos }^2(x)-1)}^{\frac{1}{2}(\frac{4am+4\sqrt{-a}{n}^2+4an-4\sqrt{-a}n+4(-a{)}^{3/2}+8\sqrt{-a}a+\sqrt{-a}+4m{n}^2-4mn+m+4n^3-4n^2+n}{8a+8n^2-8n+2}+\frac{1}{2}(-\sqrt{-a}+m+n)+\frac{1}{2}(-2m-1)+1)-\frac{1}{4}}{}_2{F}_1(\frac{1}{2}(m+n-\sqrt{-a}),\frac{4n^3+4m{n}^2+4\sqrt{-a}{n}^2-4n^2+4an-4mn-4\sqrt{-a}n+n+4(-a{)}^{3/2}+8\sqrt{-a}a+4am+m+\sqrt{-a}}{8n^2-8n+8a+2};\frac{1}{2}(2m+1);{\cos }^2(x))}{\sqrt{\cos (x)}}\}\right\}$$

Maple: cpu = 0.140 (sec), leaf count = 105 $$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{(\cos \left(x\right))}^m{(\sin \left(x\right))}^n{}_2\text{F}{}_1(\frac{n}{2}+\frac{m}{2}+\frac{i}{2}\sqrt{a},\frac{n}{2}+\frac{m}{2}-\frac{i}{2}\sqrt{a};\frac{1}{2}+m;{(\cos \left(x\right))}^2)+\mathit{\_}\mathit{C}\mathit{2}{(\cos \left(x\right))}^{-m+1}{(\sin \left(x\right))}^n{}_2\text{F}{}_1(\frac{n}{2}-\frac{m}{2}+\frac{i}{2}\sqrt{a}+\frac{1}{2},\frac{n}{2}-\frac{m}{2}-\frac{i}{2}\sqrt{a}+\frac{1}{2};\frac{3}{2}-m;{(\cos \left(x\right))}^2)\}$$