55.25.38 problem 38

Internal problem ID [13747]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.4-2.
Problem number : 38
Date solved : Thursday, October 02, 2025 at 07:55:55 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }&=-n y^{2}+a \left (1+2 n \right ) {\mathrm e}^{x} y+b y-a^{2} n \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}+c \end{align*}

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE -(-a**2*n*exp(2*x) - a*b*exp(x) - c + (2*a*n*exp(x) + a*exp(x) + b - n*y(x))*y(x))/y(x) + Derivative(y(x), x) cannot be solved by the factorable group method