Internal problem ID [9219]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1640.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+a y \left (y^{\prime }\right )^{2}+b y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.019 (sec). Leaf size: 70
dsolve(diff(diff(y(x),x),x)+a*y(x)*diff(y(x),x)^2+b*y(x)=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}\frac {a}{\sqrt {a \left ({\mathrm e}^{-\textit {\_a}^{2} a} c_{1} a -b \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {a}{\sqrt {a \left ({\mathrm e}^{-\textit {\_a}^{2} a} c_{1} a -b \right )}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.453 (sec). Leaf size: 96
DSolve[b*y[x] + a*y[x]*y'[x]^2 + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[1]^2}-b}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[2]^2}-b}}dK[2]\&\right ][x+c_2] \\ \end{align*}