Internal problem ID [9372]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1794 (book 6.203).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {a y \left (y-1\right ) y^{\prime \prime }-\left (a -1\right ) \left (2 y-1\right ) \left (y^{\prime }\right )^{2}+f y \left (y-1\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 48
dsolve(a*y(x)*(-1+y(x))*diff(diff(y(x),x),x)-(a-1)*(2*y(x)-1)*diff(y(x),x)^2+f*y(x)*(-1+y(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 1 \\ y \relax (x ) = 0 \\ c_{1} {\mathrm e}^{-\frac {f x}{a}}-c_{2}+\int _{}^{y \relax (x )}\frac {\left (\textit {\_a} \left (\textit {\_a} -1\right )\right )^{\frac {1}{a}}}{\textit {\_a} \left (\textit {\_a} -1\right )}d \textit {\_a} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.12 (sec). Leaf size: 83
DSolve[f[x]*(-1 + y[x])*y[x]*y'[x] - (-1 + a)*(-1 + 2*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 [a \text {$\#$1}^{-1/a} (-((\text {$\#$1}-1) \text {$\#$1}))^{\frac {1}{a}} \, _2F_1\left (\frac {1}{a},\frac {a-1}{a};1+\frac {1}{a};1-\text {$\#$1}\right )\&\right ]\left [\int _1^x\exp \left (-\int _1^{K[3]}\frac {f(K[1])}{a}dK[1]\right ) c_1dK[3]+c_2\right ] \\ \end{align*}