Internal problem ID [9963]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1640 (6.50).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {y^{\prime \prime }+a y {y^{\prime }}^{2}+y b=0} \]
✓ Solution by Maple
Time used: 0.015 (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*} a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {a \left ({\mathrm e}^{-a \,\textit {\_a}^{2}} c_{1} a -b \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {a \left ({\mathrm e}^{-a \,\textit {\_a}^{2}} c_{1} a -b \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.909 (sec). Leaf size: 290
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 {$\#$1}}-\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[1]^2}-b}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[2]^2}-b}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {a}}{\sqrt {e^{2 a (-c_1)-a K[1]^2}-b}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[1]^2}-b}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {a}}{\sqrt {e^{2 a (-c_1)-a K[2]^2}-b}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[2]^2}-b}}dK[2]\&\right ][x+c_2] \\ \end{align*}