Internal problem ID [9291]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1713 (book 6.122).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y^{\prime \prime } y-\left (y^{\prime }\right )^{2}-f \relax (x ) y y^{\prime }-g \relax (x ) y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 37
dsolve(diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2-f(x)*y(x)*diff(y(x),x)-g(x)*y(x)^2=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{c_{1} \left (\int {\mathrm e}^{\int f \relax (x )d x}d x \right )} {\mathrm e}^{\int {\mathrm e}^{\int f \relax (x )d x} \left (\int {\mathrm e}^{\int -f \relax (x )d x} g \relax (x )d x \right )d x} c_{2} \]
✓ Solution by Mathematica
Time used: 0.047 (sec). Leaf size: 61
DSolve[-(g[x]*y[x]^2) - f[x]*y[x]*y'[x] - y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \exp \left (\int _1^x\exp \left (\int _1^{K[3]}f(K[1])dK[1]\right ) \left (c_1+\int _1^{K[3]}\exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right ) g(K[2])dK[2]\right )dK[3]\right ) \\ \end{align*}