Internal problem ID [9024]
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.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (2 f \relax (x ) g^{\prime }\relax (x )^{2} g \relax (x )-\left (g \relax (x )^{2}-1\right ) \left (f \relax (x ) g^{\prime \prime }\relax (x )+2 f^{\prime }\relax (x ) g^{\prime }\relax (x )\right )\right ) y^{\prime }}{f \relax (x ) g^{\prime }\relax (x ) \left (g \relax (x )^{2}-1\right )}+\frac {\left (\left (g \relax (x )^{2}-1\right ) \left (f^{\prime }\relax (x ) \left (f \relax (x ) g^{\prime \prime }\relax (x )+2 f^{\prime }\relax (x ) g^{\prime }\relax (x )\right )-f \relax (x ) f^{\prime \prime }\relax (x ) g^{\prime }\relax (x )\right )-\left (2 g \relax (x ) f^{\prime }\relax (x )+v \left (v +1\right ) f \relax (x ) g^{\prime }\relax (x )\right ) f \relax (x ) g^{\prime }\relax (x )^{2}\right ) y}{f \relax (x )^{2} g^{\prime }\relax (x ) \left (g \relax (x )^{2}-1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.074 (sec). Leaf size: 21
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)
\[ y \relax (x ) = c_{1} \LegendreP \left (v , g \relax (x )\right ) f \relax (x )+c_{2} \LegendreQ \left (v , g \relax (x )\right ) f \relax (x ) \]
✓ Solution by Mathematica
Time used: 0.105 (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]
\begin{align*} y(x)\to f(x) (c_1 P_v(g(x))+c_2 Q_v(g(x))) \\ \end{align*}