60.7.174 problem 1794 (book 6.203)

Internal problem ID [11724]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1794 (book 6.203)
Date solved : Sunday, March 30, 2025 at 08:44:45 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} a y \left (-1+y\right ) y^{\prime \prime }-\left (a -1\right ) \left (2 y-1\right ) {y^{\prime }}^{2}+f y \left (-1+y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 48

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

\begin{align*} y &= 1 \\ y &= 0 \\ c_1 \,{\mathrm e}^{-\frac {f x}{a}}-c_2 +\int _{}^{y}\frac {\left (\textit {\_a} \left (-1+\textit {\_a} \right )\right )^{\frac {1}{a}}}{\textit {\_a} \left (-1+\textit {\_a} \right )}d \textit {\_a} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 1.224 (sec). Leaf size: 106

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

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

✗ Sympy

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

NotImplementedError : The given ODE -(f*y(x)**2 - f*y(x) + sqrt((8*a**2*y(x)**2*Derivative(y(x), (x, 2)) - 12*a**2*y(x)*Derivative(y(x), (x, 2)) + 4*a**2*Derivative(y(x), (x, 2)) - 8*a*y(x)**2*Derivative(y(x), (x, 2)) + 12*a*y(x)*Derivative(y(x), (x, 2)) - 4*a*Derivative(y(x), (x, 2)) + f**2*y(x)**3 - 2*f**2*y(x)**2 + f**2*y(x))*y(x)))/(2*(2*a*y(x) - a - 2*y(x) + 1)) + Derivative(y(x), x) cannot be solved by the factorable group method