2.1711   ODE No. 1711

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

$$y(x{)}^2\log (y(x))({\cos }^2(x)-n^2{\cot }^2(x))+y(x)y^{″}(x)-y^{\prime }(x{)}^2+y(x)y^{\prime }(x)(\tan (x)+\cot (x))=0$$ ✓ Mathematica : cpu = 98.9686 (sec), leaf count = 916

$$\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[3])-1}\right)\cos (K[3]){(2{\cos }^2(K[3])-2)}^{\frac{n+1}{2}}\sin (K[3]){({\cos }^2(K[3])-1)}^{\frac{1}{2}(-n-1)-\frac{1}{2}}}{\sqrt{1-{\cos }^2(K[3])}}+\frac{(-1{)}^{1-n}{4}^{\frac{n+1}{2}-\frac{1}{2}}{K}_n\left(\sqrt{{\cos }^2(K[3])-1}\right)\cos (K[3]){(2{\cos }^2(K[3])-2)}^{\frac{n+1}{2}}\sin (K[3]){({\cos }^2(K[3])-1)}^{\frac{1}{2}(-n-1)+\frac{1}{2}}}{{(1-{\cos }^2(K[3]))}^{3/2}}+\frac{(-1{)}^{1-n}{2}^{n+1}(n+1)K_n\left(\sqrt{{\cos }^2(K[3])-1}\right)\cos (K[3]){(2{\cos }^2(K[3])-2)}^{\frac{n+1}{2}-1}\sin (K[3]){({\cos }^2(K[3])-1)}^{\frac{1}{2}(-n-1)+\frac{1}{2}}}{\sqrt{1-{\cos }^2(K[3])}}+\frac{{(-\frac{1}{2})}^{1-n}(-K_{n-1}\left(\sqrt{{\cos }^2(K[3])-1}\right)-K_{n+1}\left(\sqrt{{\cos }^2(K[3])-1}\right))\cos (K[3]){(2{\cos }^2(K[3])-2)}^{\frac{n+1}{2}}\sin (K[3]){({\cos }^2(K[3])-1)}^{\frac{1}{2}(-n-1)}}{\sqrt{1-{\cos }^2(K[3])}}+c_1(-\frac{2^{n+1}(\frac{1}{2}(-n-1)+\frac{1}{2})I_n\left(\sqrt{{\cos }^2(K[3])-1}\right)\cos (K[3]){(2{\cos }^2(K[3])-2)}^{\frac{n+1}{2}}\sin (K[3]){({\cos }^2(K[3])-1)}^{\frac{1}{2}(-n-1)-\frac{1}{2}}}{\sqrt{1-{\cos }^2(K[3])}}-\frac{4^{\frac{n+1}{2}-\frac{1}{2}}{I}_n\left(\sqrt{{\cos }^2(K[3])-1}\right)\cos (K[3]){(2{\cos }^2(K[3])-2)}^{\frac{n+1}{2}}\sin (K[3]){({\cos }^2(K[3])-1)}^{\frac{1}{2}(-n-1)+\frac{1}{2}}}{{(1-{\cos }^2(K[3]))}^{3/2}}-\frac{2^{n+1}(n+1)I_n\left(\sqrt{{\cos }^2(K[3])-1}\right)\cos (K[3]){(2{\cos }^2(K[3])-2)}^{\frac{n+1}{2}-1}\sin (K[3]){({\cos }^2(K[3])-1)}^{\frac{1}{2}(-n-1)+\frac{1}{2}}}{\sqrt{1-{\cos }^2(K[3])}}-\frac{2^{n-1}(I_{n-1}\left(\sqrt{{\cos }^2(K[3])-1}\right)+I_{n+1}\left(\sqrt{{\cos }^2(K[3])-1}\right))\cos (K[3]){(2{\cos }^2(K[3])-2)}^{\frac{n+1}{2}}\sin (K[3]){({\cos }^2(K[3])-1)}^{\frac{1}{2}(-n-1)}}{\sqrt{1-{\cos }^2(K[3])}})}{\frac{(-1{)}^{1-n}{4}^{\frac{n+1}{2}-\frac{1}{2}}{K}_n\left(\sqrt{{\cos }^2(K[3])-1}\right){({\cos }^2(K[3])-1)}^{\frac{1}{2}(-n-1)+\frac{1}{2}}{(2{\cos }^2(K[3])-2)}^{\frac{n+1}{2}}}{\sqrt{1-{\cos }^2(K[3])}}-\frac{4^{\frac{n+1}{2}-\frac{1}{2}}{I}_n\left(\sqrt{{\cos }^2(K[3])-1}\right)c_1{({\cos }^2(K[3])-1)}^{\frac{1}{2}(-n-1)+\frac{1}{2}}{(2{\cos }^2(K[3])-2)}^{\frac{n+1}{2}}}{\sqrt{1-{\cos }^2(K[3])}}}dK[3]}}\}\right\}$$ ✓ Maple : cpu = 0.709 (sec), leaf count = 81

$$\{y\left(x\right)={\mathrm{e}}^{\frac{c_2\mathrm{BesselY}(n,\sin \left(x\right))}{(\mathrm{BesselJ}(n,\sin \left(x\right))\mathrm{BesselY}(n+1,\sin \left(x\right))-\mathrm{BesselJ}(n+1,\sin \left(x\right))\mathrm{BesselY}(n,\sin \left(x\right)))\sin \left(x\right)}}{\mathrm{e}}^{-\frac{c_1\mathrm{BesselJ}(n,\sin \left(x\right))}{(\mathrm{BesselJ}(n,\sin \left(x\right))\mathrm{BesselY}(n+1,\sin \left(x\right))-\mathrm{BesselJ}(n+1,\sin \left(x\right))\mathrm{BesselY}(n,\sin \left(x\right)))\sin \left(x\right)}}\}$$