Internal problem ID [8665]
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.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\left (\frac {g^{\prime \prime }\relax (x )}{g^{\prime }\relax (x )}+\frac {\left (2 v -1\right ) g^{\prime }\relax (x )}{g \relax (x )}+\frac {2 h^{\prime }\relax (x )}{h \relax (x )}\right ) y^{\prime }+\left (\frac {h^{\prime }\relax (x ) \left (\frac {g^{\prime \prime }\relax (x )}{g^{\prime }\relax (x )}+\frac {\left (2 v -1\right ) g^{\prime }\relax (x )}{g \relax (x )}+\frac {2 h^{\prime }\relax (x )}{h \relax (x )}\right )}{h \relax (x )}-\frac {h^{\prime \prime }\relax (x )}{h \relax (x )}+g^{\prime }\relax (x )^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 29
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)
\[ y \relax (x ) = c_{1} \BesselJ \left (v , g \relax (x )\right ) h \relax (x ) g \relax (x )^{v}+c_{2} \BesselY \left (v , g \relax (x )\right ) h \relax (x ) g \relax (x )^{v} \]
✓ Solution by Mathematica
Time used: 0.101 (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]
\begin{align*} y(x)\to h(x) g(x)^v (c_1 J_v(g(x))+c_2 Y_v(g(x))) \\ \end{align*}