Internal problem ID [9290]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1711.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y^{\prime \prime } y-\left (y^{\prime }\right )^{2}+\left (\tan \relax (x )+\cot \relax (x )\right ) y y^{\prime }+\left (\cos ^{2}\relax (x )-n^{2} \left (\cot ^{2}\relax (x )\right )\right ) y^{2} \ln \relax (y)=0} \end {gather*}
✓ Solution by Maple
Time used: 0.089 (sec). Leaf size: 81
dsolve(diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2+(tan(x)+cot(x))*y(x)*diff(y(x),x)+(cos(x)^2-n^2*cot(x)^2)*y(x)^2*ln(y(x))=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{\frac {\BesselJ \left (n , \sin \relax (x )\right ) c_{1}}{\sin \relax (x ) \left (-\BesselJ \left (n , \sin \relax (x )\right ) \BesselY \left (n +1, \sin \relax (x )\right )+\BesselJ \left (n +1, \sin \relax (x )\right ) \BesselY \left (n , \sin \relax (x )\right )\right )}} {\mathrm e}^{\frac {\BesselY \left (n , \sin \relax (x )\right ) c_{2}}{\sin \relax (x ) \left (\BesselJ \left (n , \sin \relax (x )\right ) \BesselY \left (n +1, \sin \relax (x )\right )-\BesselJ \left (n +1, \sin \relax (x )\right ) \BesselY \left (n , \sin \relax (x )\right )\right )}} \]
✓ Solution by Mathematica
Time used: 14.023 (sec). Leaf size: 858
DSolve[(Cos[x]^2 - n^2*Cot[x]^2)*Log[y[x]]*y[x]^2 + (Cot[x] + Tan[x])*y[x]*y'[x] - y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {(-1)^{-n} 2^{3 n/2} e^{-(-1)^{-n} 2^{-\frac {3 n}{2}-4} \left (c_2-\int _1^x-\frac {4 \cot (K[3]) y(K[3]) \left (2^{3 n+1} \sqrt {\cos (2 K[3])-1} \left (2 n^2+\cos (2 K[3])-1\right ) \csc (K[3]) \log (y(K[3])) K_n(i \sin (K[3])){}^2+(-1)^n 2^{\frac {3 n}{2}+2} c_1 \sec ^2(K[3]) K_n(i \sin (K[3]))-2 (-1)^{2 n} \sqrt {\cos (2 K[3])-1} \csc (K[3]) \int _1^{K[3]}\frac {(-1)^{-n} 2^{\frac {3 n}{2}-\frac {1}{2}} \left (-i K_n(i \sin (K[1])) \left (2 n^2+\cos (2 K[1])-1\right ) \cot (K[1]) \log (y(K[1])) y(K[1])-(K_{n-1}(i \sin (K[1]))+K_{n+1}(i \sin (K[1]))) \sin (K[1]) y'(K[1])\right )}{y(K[1])}dK[1]{}^2-(-1)^n c_1 \sqrt {\cos (2 K[3])-1} \left (2^{\frac {3 n}{2}+\frac {1}{2}} K_{n-1}(i \sin (K[3]))+2^{\frac {3 n}{2}+\frac {1}{2}} K_{n+1}(i \sin (K[3]))+2 (-1)^n c_1 \csc (K[3])\right )-(-1)^n \left (\sqrt {\cos (2 K[3])-1} \left (2^{\frac {3 n}{2}+\frac {1}{2}} K_{n-1}(i \sin (K[3]))+2^{\frac {3 n}{2}+\frac {1}{2}} K_{n+1}(i \sin (K[3]))+4 (-1)^n c_1 \csc (K[3])\right )-2^{\frac {3 n}{2}+2} K_n(i \sin (K[3])) \sec ^2(K[3])\right ) \int _1^{K[3]}\frac {(-1)^{-n} 2^{\frac {3 n}{2}-\frac {1}{2}} \left (-i K_n(i \sin (K[1])) \left (2 n^2+\cos (2 K[1])-1\right ) \cot (K[1]) \log (y(K[1])) y(K[1])-(K_{n-1}(i \sin (K[1]))+K_{n+1}(i \sin (K[1]))) \sin (K[1]) y'(K[1])\right )}{y(K[1])}dK[1]\right )+2^{3 n+\frac {7}{2}} K_n(i \sin (K[3])) (K_{n-1}(i \sin (K[3]))+K_{n+1}(i \sin (K[3]))) \sin (K[3]) y'(K[3])}{K_n(i \sin (K[3])) y(K[3]) \left (c_1+\int _1^{K[3]}\frac {(-1)^{-n} 2^{\frac {3 n}{2}-\frac {1}{2}} \left (-i K_n(i \sin (K[1])) \left (2 n^2+\cos (2 K[1])-1\right ) \cot (K[1]) \log (y(K[1])) y(K[1])-(K_{n-1}(i \sin (K[1]))+K_{n+1}(i \sin (K[1]))) \sin (K[1]) y'(K[1])\right )}{y(K[1])}dK[1]\right )}dK[3]\right )} K_n\left (\sqrt {-\sin ^2(x)}\right ) \sqrt {\cos (2 x)-1} \sec (x)}{c_1+\int _1^x\frac {(-1)^{-n} 2^{3 n/2} \left (\left (K_{n-1}\left (\sqrt {-\sin ^2(K[1])}\right )+K_{n+1}\left (\sqrt {-\sin ^2(K[1])}\right )\right ) \csc (K[1]) y'(K[1]) \left (-\sin ^2(K[1])\right )^{3/2}+K_n\left (\sqrt {-\sin ^2(K[1])}\right ) \cos (K[1]) \left (2 n^2+\cos (2 K[1])-1\right ) \log (y(K[1])) y(K[1])\right )}{\sqrt {\cos (2 K[1])-1} y(K[1])}dK[1]} \\ \end{align*}