54.7.175 problem 1795 (book 6.204)

Internal problem ID [13024]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1795 (book 6.204)
Date solved : Wednesday, October 01, 2025 at 02:57:18 AM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 58

ode:=a*b*y(x)*(y(x)-1)*diff(diff(y(x),x),x)-((2*a*b-a-b)*y(x)+(-a+1)*b)*diff(y(x),x)^2+f*y(x)*(y(x)-1)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 1.523 (sec). Leaf size: 120

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

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

✗ Sympy

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

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