Internal problem ID [9771]
Internal file name [OUTPUT/8714_Monday_June_06_2022_05_19_01_AM_31804215/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1445.
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, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\frac {\left (2 f \left (x \right ) {g^{\prime }\left (x \right )}^{2} g \left (x \right )-\left (g \left (x \right )^{2}-1\right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )\right ) y^{\prime }}{f \left (x \right ) g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}+\frac {\left (\left (g \left (x \right )^{2}-1\right ) \left (f^{\prime }\left (x \right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )-f \left (x \right ) f^{\prime \prime }\left (x \right ) g^{\prime }\left (x \right )\right )-\left (2 g \left (x \right ) f^{\prime }\left (x \right )+v \left (v +1\right ) f \left (x \right ) g^{\prime }\left (x \right )\right ) f \left (x \right ) {g^{\prime }\left (x \right )}^{2}\right ) y}{f \left (x \right )^{2} g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- 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 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 -> elliptic -> Legendre <- Legendre successful <- special function solution successful <- change of variables successful`
✓ Solution by Maple
Time used: 0.203 (sec). Leaf size: 20
dsolve(diff(diff(y(x),x),x) = -(2*f(x)*diff(g(x),x)^2*g(x)-(g(x)^2-1)*(f(x)*diff(diff(g(x),x),x)+2*diff(f(x),x)*diff(g(x),x)))/f(x)/diff(g(x),x)/(g(x)^2-1)*diff(y(x),x)-((g(x)^2-1)*(diff(f(x),x)*(f(x)*diff(diff(g(x),x),x)+2*diff(f(x),x)*diff(g(x),x))-f(x)*diff(diff(f(x),x),x)*diff(g(x),x))-(2*diff(f(x),x)*g(x)+v*(v+1)*f(x)*diff(g(x),x))*f(x)*diff(g(x),x)^2)/f(x)^2/diff(g(x),x)/(g(x)^2-1)*y(x),y(x), singsol=all)
\[ y \left (x \right ) = f \left (x \right ) \left (\operatorname {LegendreQ}\left (v , g \left (x \right )\right ) c_{2} +\operatorname {LegendreP}\left (v , g \left (x \right )\right ) c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.197 (sec). Leaf size: 23
DSolve[y''[x] == -((y'[x]*(2*f[x]*g[x]*Derivative[1][g][x]^2 - (-1 + g[x]^2)*(2*Derivative[1][f][x]*Derivative[1][g][x] + f[x]*Derivative[2][g][x])))/(f[x]*(-1 + g[x]^2)*Derivative[1][g][x])) - (y[x]*(-(f[x]*Derivative[1][g][x]^2*(2*g[x]*Derivative[1][f][x] + v*(1 + v)*f[x]*Derivative[1][g][x])) + (-1 + g[x]^2)*(-(f[x]*Derivative[1][g][x]*Derivative[2][f][x]) + Derivative[1][f][x]*(2*Derivative[1][f][x]*Derivative[1][g][x] + f[x]*Derivative[2][g][x]))))/(f[x]^2*(-1 + g[x]^2)*Derivative[1][g][x]),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to f(x) (c_1 \operatorname {LegendreP}(v,g(x))+c_2 \operatorname {LegendreQ}(v,g(x))) \]