8.121   ODE No. 1711

$$\boxed{{\displaystyle \left(\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)\right)y\left(x\right)-{\left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)}^2+(\tan \left(x\right)+\cot \left(x\right))y\left(x\right)\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)+({(\cos \left(x\right))}^2-n^2{(\cot \left(x\right))}^2){\left(y\left(x\right)\right)}^2\ln \left(y\left(x\right)\right)=0}}$$

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

Mathematica: cpu = 635.571707 (sec), leaf count = 915 $$\left\{\{y(x)\to e^{e^{c_2+{\int }_1^x-\frac{\frac{(-1{)}^{1-n}{2}^{n+1}(\frac{1}{2}(-n-1)+\frac{1}{2})K_n\left(\sqrt{{\cos }^2(K[2])-1}\right)\cos (K[2]){(2{\cos }^2(K[2])-2)}^{\frac{n+1}{2}}\sin (K[2]){({\cos }^2(K[2])-1)}^{\frac{1}{2}(-n-1)-\frac{1}{2}}}{\sqrt{1-{\cos }^2(K[2])}}+\frac{(-1{)}^{1-n}{4}^{\frac{n+1}{2}-\frac{1}{2}}{K}_n\left(\sqrt{{\cos }^2(K[2])-1}\right)\cos (K[2]){(2{\cos }^2(K[2])-2)}^{\frac{n+1}{2}}\sin (K[2]){({\cos }^2(K[2])-1)}^{\frac{1}{2}(-n-1)+\frac{1}{2}}}{{(1-{\cos }^2(K[2]))}^{3/2}}+\frac{(-1{)}^{1-n}{2}^{n+1}(n+1)K_n\left(\sqrt{{\cos }^2(K[2])-1}\right)\cos (K[2]){(2{\cos }^2(K[2])-2)}^{\frac{n+1}{2}-1}\sin (K[2]){({\cos }^2(K[2])-1)}^{\frac{1}{2}(-n-1)+\frac{1}{2}}}{\sqrt{1-{\cos }^2(K[2])}}+\frac{{(-\frac{1}{2})}^{1-n}(-K_{n-1}\left(\sqrt{{\cos }^2(K[2])-1}\right)-K_{n+1}\left(\sqrt{{\cos }^2(K[2])-1}\right))\cos (K[2]){(2{\cos }^2(K[2])-2)}^{\frac{n+1}{2}}\sin (K[2]){({\cos }^2(K[2])-1)}^{\frac{1}{2}(-n-1)}}{\sqrt{1-{\cos }^2(K[2])}}+c_1(-\frac{2^{n+1}(\frac{1}{2}(-n-1)+\frac{1}{2})I_n\left(\sqrt{{\cos }^2(K[2])-1}\right)\cos (K[2]){(2{\cos }^2(K[2])-2)}^{\frac{n+1}{2}}\sin (K[2]){({\cos }^2(K[2])-1)}^{\frac{1}{2}(-n-1)-\frac{1}{2}}}{\sqrt{1-{\cos }^2(K[2])}}-\frac{4^{\frac{n+1}{2}-\frac{1}{2}}{I}_n\left(\sqrt{{\cos }^2(K[2])-1}\right)\cos (K[2]){(2{\cos }^2(K[2])-2)}^{\frac{n+1}{2}}\sin (K[2]){({\cos }^2(K[2])-1)}^{\frac{1}{2}(-n-1)+\frac{1}{2}}}{{(1-{\cos }^2(K[2]))}^{3/2}}-\frac{2^{n+1}(n+1)I_n\left(\sqrt{{\cos }^2(K[2])-1}\right)\cos (K[2]){(2{\cos }^2(K[2])-2)}^{\frac{n+1}{2}-1}\sin (K[2]){({\cos }^2(K[2])-1)}^{\frac{1}{2}(-n-1)+\frac{1}{2}}}{\sqrt{1-{\cos }^2(K[2])}}-\frac{2^{n-1}(I_{n-1}\left(\sqrt{{\cos }^2(K[2])-1}\right)+I_{n+1}\left(\sqrt{{\cos }^2(K[2])-1}\right))\cos (K[2]){(2{\cos }^2(K[2])-2)}^{\frac{n+1}{2}}\sin (K[2]){({\cos }^2(K[2])-1)}^{\frac{1}{2}(-n-1)}}{\sqrt{1-{\cos }^2(K[2])}})}{\frac{(-1{)}^{1-n}{4}^{\frac{n+1}{2}-\frac{1}{2}}{K}_n\left(\sqrt{{\cos }^2(K[2])-1}\right){({\cos }^2(K[2])-1)}^{\frac{1}{2}(-n-1)+\frac{1}{2}}{(2{\cos }^2(K[2])-2)}^{\frac{n+1}{2}}}{\sqrt{1-{\cos }^2(K[2])}}-\frac{4^{\frac{n+1}{2}-\frac{1}{2}}{I}_n\left(\sqrt{{\cos }^2(K[2])-1}\right)c_1{({\cos }^2(K[2])-1)}^{\frac{1}{2}(-n-1)+\frac{1}{2}}{(2{\cos }^2(K[2])-2)}^{\frac{n+1}{2}}}{\sqrt{1-{\cos }^2(K[2])}}}dK[2]}}\}\right\}$$

Maple: cpu = 0.390 (sec), leaf count = 81 $$\{y\left(x\right)=1{\mathrm{e}}^{\frac{{\mathit{Y}}_n(\sin \left(x\right))\mathit{\_}\mathit{C}\mathit{2}}{\sin \left(x\right)({\mathit{J}}_n(\sin \left(x\right)){\mathit{Y}}_{n+1}(\sin \left(x\right))-{\mathit{J}}_{n+1}(\sin \left(x\right)){\mathit{Y}}_n(\sin \left(x\right)))}}{\left({\mathrm{e}}^{\frac{{\mathit{J}}_n(\sin \left(x\right))\mathit{\_}\mathit{C}\mathit{1}}{\sin \left(x\right)({\mathit{J}}_n(\sin \left(x\right)){\mathit{Y}}_{n+1}(\sin \left(x\right))-{\mathit{J}}_{n+1}(\sin \left(x\right)){\mathit{Y}}_n(\sin \left(x\right)))}}\right)}^{-1}\}$$