Internal problem ID [8664]
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.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\left (\frac {2 f^{\prime }\relax (x )}{f \relax (x )}+\frac {g^{\prime \prime }\relax (x )}{g^{\prime }\relax (x )}-\frac {g^{\prime }\relax (x )}{g \relax (x )}\right ) y^{\prime }+\left (\frac {f^{\prime }\relax (x ) \left (\frac {2 f^{\prime }\relax (x )}{f \relax (x )}+\frac {g^{\prime \prime }\relax (x )}{g^{\prime }\relax (x )}-\frac {g^{\prime }\relax (x )}{g \relax (x )}\right )}{f \relax (x )}-\frac {f^{\prime \prime }\relax (x )}{f \relax (x )}-\frac {g^{\prime }\relax (x )^{2} v^{2}}{g \relax (x )^{2}}+g^{\prime }\relax (x )^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 21
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)
\[ y \relax (x ) = c_{1} \BesselJ \left (v , g \relax (x )\right ) f \relax (x )+c_{2} \BesselY \left (v , g \relax (x )\right ) f \relax (x ) \]
✓ Solution by Mathematica
Time used: 0.107 (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]
\begin{align*} y(x)\to f(x) \left (c_1 J_{\sqrt {v^2}}(g(x))+c_2 Y_{\sqrt {v^2}}(g(x))\right ) \\ \end{align*}