Internal problem ID [10033]
Internal file name [OUTPUT/8980_Monday_June_06_2022_06_07_18_AM_87135055/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1712 (book 6.121).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}+\left (\tan \left (x \right )+\cot \left (x \right )\right ) y y^{\prime }+\left (\cos \left (x \right )^{2}-n^{2} \cot \left (x \right )^{2}\right ) y^{2} \ln \left (y\right )=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful Change of variables used: [x = arcsin(t)] Linear ODE actually solved: (2*n^2*t^4-2*t^6-4*n^2*t^2+4*t^4+2*n^2+2*t^2-2)*u(t)+(-2*t^5+2*t)*diff(u(t),t)+(-2*t^6+4*t^4-2*t^2)*diff(diff(u(t),t),t) = 0 <- change of variables successful <- 2nd order, 2 integrating factors of the form mu(x,y) successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 22
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)
\[ y \left (x \right ) = {\mathrm e}^{\frac {\pi \left (\operatorname {BesselJ}\left (n , \sin \left (x \right )\right ) c_{1} -c_{2} \operatorname {BesselY}\left (n , \sin \left (x \right )\right )\right )}{2}} \]
✓ Solution by Mathematica
Time used: 81.947 (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]
\[ 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]} \]