Internal problem ID [9365]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1786.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {2 y \left (1-y\right ) y^{\prime \prime }-\left (1-2 y\right ) \left (y^{\prime }\right )^{2}+y \left (1-y\right ) y^{\prime } f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.086 (sec). Leaf size: 59
dsolve(2*y(x)*(1-y(x))*diff(diff(y(x),x),x)-(1-2*y(x))*diff(y(x),x)^2+y(x)*(1-y(x))*diff(y(x),x)*f(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (4 \,{\mathrm e}^{\int 2 c_{1} {\mathrm e}^{\int -\frac {f \relax (x )}{2}d x}d x} c_{2}^{2}+4 \,{\mathrm e}^{c_{1} \left (\int {\mathrm e}^{-\frac {\left (\int f \relax (x )d x \right )}{2}}d x \right )} c_{2}+1\right ) {\mathrm e}^{\int -c_{1} {\mathrm e}^{\int -\frac {f \relax (x )}{2}d x}d x}}{8 c_{2}} \]
✓ Solution by Mathematica
Time used: 0.079 (sec). Leaf size: 45
DSolve[f[x]*(1 - y[x])*y[x]*y'[x] - (1 - 2*y[x])*y'[x]^2 + 2*(1 - y[x])*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sin ^2\left (\frac {1}{2} \left (\int _1^x-\exp \left (-\int _1^{K[3]}\frac {1}{2} f(K[1])dK[1]\right ) c_1dK[3]+c_2\right )\right ) \\ \end{align*}