Internal problem ID [9413]
Internal file name [OUTPUT/8353_Monday_June_06_2022_02_49_17_AM_41422348/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1084.
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.
\[ \boxed {y^{\prime \prime }-\left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right ) y^{\prime }+\left (\frac {f^{\prime }\left (x \right ) \left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right )}{f \left (x \right )}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}-\frac {v^{2} {g^{\prime }\left (x \right )}^{2}}{g \left (x \right )^{2}}+{g^{\prime }\left (x \right )}^{2}\right ) y=0} \]
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: 0.093 (sec). Leaf size: 20
dsolve(diff(diff(y(x),x),x)-(2*diff(f(x),x)/f(x)+diff(diff(g(x),x),x)/diff(g(x),x)-diff(g(x),x)/g(x))*diff(y(x),x)+(diff(f(x),x)/f(x)*(2*diff(f(x),x)/f(x)+diff(diff(g(x),x),x)/diff(g(x),x)-diff(g(x),x)/g(x))-diff(diff(f(x),x),x)/f(x)-v^2*diff(g(x),x)^2/g(x)^2+diff(g(x),x)^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = f \left (x \right ) \left (\operatorname {BesselJ}\left (v , g \left (x \right )\right ) c_{1} +\operatorname {BesselY}\left (v , g \left (x \right )\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.216 (sec). Leaf size: 35
DSolve[-(y'[x]*((2*Derivative[1][f][x])/f[x] - Derivative[1][g][x]/g[x] + Derivative[2][g][x]/Derivative[1][g][x])) + y[x]*(Derivative[1][g][x]^2 - (v^2*Derivative[1][g][x]^2)/g[x]^2 - Derivative[2][f][x]/f[x] + (Derivative[1][f][x]*((2*Derivative[1][f][x])/f[x] - Derivative[1][g][x]/g[x] + Derivative[2][g][x]/Derivative[1][g][x]))/f[x]) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to f(x) \left (c_1 \operatorname {BesselJ}\left (\sqrt {v^2},g(x)\right )+c_2 \operatorname {BesselY}\left (\sqrt {v^2},g(x)\right )\right ) \]