Internal problem ID [9216]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1641 (6.51).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {y^{\prime \prime }+f \left (y\right ) {y^{\prime }}^{2}+g \left (x \right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 29
dsolve(diff(diff(y(x),x),x)+f(y(x))*diff(y(x),x)^2+g(x)*diff(y(x),x)=0,y(x), singsol=all)
\[ \int _{}^{y \left (x \right )}{\mathrm e}^{\int f \left (\textit {\_b} \right )d \textit {\_b}}d \textit {\_b} -c_{1} \left (\int {\mathrm e}^{-\left (\int g \left (x \right )d x \right )}d x \right )-c_{2} = 0 \]
✓ Solution by Mathematica
Time used: 0.033 (sec). Leaf size: 61
DSolve[g[x]*y'[x] + f[y[x]]*y'[x]^2 + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\exp \left (-\int _1^{K[4]}-f(K[1])dK[1]\right )dK[4]\&\right ]\left [\int _1^x-\exp \left (-\int _1^{K[5]}g(K[2])dK[2]\right ) c_1dK[5]+c_2\right ] \\ \end{align*}