Internal problem ID [10108]
Internal file name [OUTPUT/9055_Monday_June_06_2022_06_18_32_AM_81130002/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1787 (book 6.196).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {2 y \left (1-y\right ) y^{\prime \prime }-\left (1-2 y\right ) {y^{\prime }}^{2}+y \left (1-y\right ) y^{\prime } f \left (x \right )=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville <- 2nd_order Liouville successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 43
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)
\[ y \left (x \right ) = \frac {4 \,{\mathrm e}^{c_{1} \left (\int {\mathrm e}^{-\frac {\left (\int f \left (x \right )d x \right )}{2}}d x \right )} c_{2}^{2}+4 c_{2} +{\mathrm e}^{-c_{1} \left (\int {\mathrm e}^{-\frac {\left (\int f \left (x \right )d x \right )}{2}}d x \right )}}{8 c_{2}} \]
✓ Solution by Mathematica
Time used: 0.138 (sec). Leaf size: 91
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]
\[ y(x)\to \frac {1}{4} \exp \left (-i \left (\int _1^x-\exp \left (-\int _1^{K[1]}\frac {1}{2} f(K[1])dK[1]\right ) c_1dK[1]+c_2\right )\right ) \left (1+\exp \left (i \left (\int _1^x-\exp \left (-\int _1^{K[1]}\frac {1}{2} f(K[1])dK[1]\right ) c_1dK[1]+c_2\right )\right )\right ){}^2 \]