60.7.196 problem 1787 (book 6.196)

Internal problem ID [11785]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1787 (book 6.196)
Date solved : Tuesday, January 28, 2025 at 06:11:25 PM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 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 \end{align*}

✓ Solution by Maple

Time used: 0.701 (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)
 

\[ y = \frac {4 \,{\mathrm e}^{c_{1} \left (\int {\mathrm e}^{-\frac {\left (\int fd x \right )}{2}}d x \right )} c_{2}^{2}+4 c_{2} +{\mathrm e}^{-c_{1} \left (\int {\mathrm e}^{-\frac {\left (\int fd x \right )}{2}}d x \right )}}{8 c_{2}} \]

✓ Solution by Mathematica

Time used: 1.119 (sec). Leaf size: 81

DSolve[f[x]*(1 - y[x])*y[x]*D[y[x],x] - (1 - 2*y[x])*D[y[x],x]^2 + 2*(1 - y[x])*y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\exp \left (-\int _1^{K[1]}\left (\frac {1}{K[1]-1}-\frac {1}{2 (K[1]-1) K[1]}\right )dK[1]\right )dK[1]\&\right ]\left [\int _1^x-\exp \left (-\int _1^{K[2]}\frac {1}{2} f(K[2])dK[2]\right ) c_1dK[2]+c_2\right ] \]