Internal problem ID [9371]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1793 (book 6.202).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {a y \left (y-1\right ) y^{\prime \prime }+\left (b y+c \right ) \left (y^{\prime }\right )^{2}+h \relax (y)=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 192
dsolve(a*y(x)*(-1+y(x))*diff(diff(y(x),x),x)+(b*y(x)+c)*diff(y(x),x)^2+h(y(x))=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}\frac {a \,\textit {\_b}^{-\frac {c}{a}} \left (\textit {\_b} -1\right )^{\frac {c +b}{a}}}{\sqrt {-a \left (-c_{1} a +2 \left (\int \frac {\left (\textit {\_b} -1\right )^{\frac {2 b}{a}} \left (\textit {\_b} -1\right )^{\frac {2 c}{a}} \textit {\_b}^{-\frac {2 c}{a}} h \left (\textit {\_b} \right )}{\textit {\_b} \left (\textit {\_b} -1\right )}d \textit {\_b} \right )\right )}}d \textit {\_b} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {a \,\textit {\_b}^{-\frac {c}{a}} \left (\textit {\_b} -1\right )^{\frac {c +b}{a}}}{\sqrt {-a \left (-c_{1} a +2 \left (\int \frac {\left (\textit {\_b} -1\right )^{\frac {2 b}{a}} \left (\textit {\_b} -1\right )^{\frac {2 c}{a}} \textit {\_b}^{-\frac {2 c}{a}} h \left (\textit {\_b} \right )}{\textit {\_b} \left (\textit {\_b} -1\right )}d \textit {\_b} \right )\right )}}d \textit {\_b} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.75 (sec). Leaf size: 232
DSolve[h[y[x]] + (c + b*y[x])*y'[x]^2 + a*(-1 + y[x])*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {(1-K[2])^{\frac {1}{2} \left (\frac {2 b}{a}+\frac {2 c}{a}\right )} K[2]^{-\frac {c}{a}}}{\sqrt {c_1+2 \int _1^{K[2]}-\frac {\exp \left (-2 \left (\frac {c \log (K[1])}{a}-\frac {(b+c) \log (1-K[1])}{a}\right )\right ) h(K[1])}{a (K[1]-1) K[1]}dK[1]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {(1-K[3])^{\frac {1}{2} \left (\frac {2 b}{a}+\frac {2 c}{a}\right )} K[3]^{-\frac {c}{a}}}{\sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp \left (-2 \left (\frac {c \log (K[1])}{a}-\frac {(b+c) \log (1-K[1])}{a}\right )\right ) h(K[1])}{a (K[1]-1) K[1]}dK[1]}}dK[3]\&\right ][x+c_2] \\ \end{align*}