Internal problem ID [9414]
Internal file name [OUTPUT/8354_Monday_June_06_2022_02_49_26_AM_17203197/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1085.
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.
\[ \boxed {y^{\prime \prime }-\left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right ) y^{\prime }+\left (\frac {h^{\prime }\left (x \right ) \left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right )}{h \left (x \right )}-\frac {h^{\prime \prime }\left (x \right )}{h \left (x \right )}+{g^{\prime }\left (x \right )}^{2}\right ) y=0} \]
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 <- Bessel successful <- special function solution successful <- change of variables successful`
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 24
dsolve(diff(diff(y(x),x),x)-(diff(diff(g(x),x),x)/diff(g(x),x)+(2*v-1)*diff(g(x),x)/g(x)+2*diff(h(x),x)/h(x))*diff(y(x),x)+(diff(h(x),x)/h(x)*(diff(diff(g(x),x),x)/diff(g(x),x)+(2*v-1)*diff(g(x),x)/g(x)+2*diff(h(x),x)/h(x))-diff(diff(h(x),x),x)/h(x)+diff(g(x),x)^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = g \left (x \right )^{v} h \left (x \right ) \left (\operatorname {BesselJ}\left (v , g \left (x \right )\right ) c_{1} +\operatorname {BesselY}\left (v , g \left (x \right )\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.2 (sec). Leaf size: 27
DSolve[-(y'[x]*(((-1 + 2*v)*Derivative[1][g][x])/g[x] + (2*Derivative[1][h][x])/h[x] + Derivative[2][g][x]/Derivative[1][g][x])) + y[x]*(Derivative[1][g][x]^2 + (Derivative[1][h][x]*(((-1 + 2*v)*Derivative[1][g][x])/g[x] + (2*Derivative[1][h][x])/h[x] + Derivative[2][g][x]/Derivative[1][g][x]))/h[x] - Derivative[2][h][x]/h[x]) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to h(x) g(x)^v (c_1 \operatorname {BesselJ}(v,g(x))+c_2 \operatorname {BesselY}(v,g(x))) \]