60.7.169 problem 1787 (book 6.196)

Internal problem ID [11719]
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 : Sunday, March 30, 2025 at 08:44:33 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*}

✓ Maple. Time used: 0.082 (sec). Leaf size: 43

ode:=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; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {4 \,{\mathrm e}^{c_1 \int {\mathrm e}^{-\frac {\int f \left (x \right )d x}{2}}d x} c_2^{2}+4 c_2 +{\mathrm e}^{-c_1 \int {\mathrm e}^{-\frac {\int f \left (x \right )d x}{2}}d x}}{8 c_2} \]

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

ode=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; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
f = Function("f") 
ode = Eq((1 - y(x))*f(x)*y(x)*Derivative(y(x), x) + (2 - 2*y(x))*y(x)*Derivative(y(x), (x, 2)) + (2*y(x) - 1)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(sqrt((y(x) - 1)*(f(x)**2*y(x)**2 - f(x)**2*y(x) + 16*y(x)*Derivative(y(x), (x, 2)) - 8*Derivative(y(x), (x, 2)))*y(x)) + (y(x) - 1)*f(x)*y(x))/(2*(2*y(x) - 1)) + Derivative(y(x), x) cannot be solved by the factorable group method