55.26.25 problem 25

Internal problem ID [13772]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.4. Equations Containing Polynomial Functions of y. subsection 1.4.1-2 Abel equations of the first kind.
Problem number : 25
Date solved : Thursday, October 02, 2025 at 08:06:28 AM
CAS classification : [_Abel]

\begin{align*} y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{3}+3 a b \,{\mathrm e}^{\lambda x} y^{2}+c y-2 a \,b^{3} {\mathrm e}^{\lambda x}+b c \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 128

ode:=diff(y(x),x) = a*exp(lambda*x)*y(x)^3+3*a*b*exp(lambda*x)*y(x)^2+c*y(x)-2*a*b^3*exp(lambda*x)+b*c; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -b \\ y &= -\frac {{\mathrm e}^{x c -\frac {3 a \,b^{2} {\mathrm e}^{\lambda x}}{\lambda }}}{\sqrt {c_1 -2 a \int {\mathrm e}^{\frac {-6 \,{\mathrm e}^{\lambda x} a \,b^{2}+2 \lambda \left (c +\frac {\lambda }{2}\right ) x}{\lambda }}d x}}-b \\ y &= \frac {{\mathrm e}^{x c -\frac {3 a \,b^{2} {\mathrm e}^{\lambda x}}{\lambda }}}{\sqrt {c_1 -2 a \int {\mathrm e}^{\frac {-6 \,{\mathrm e}^{\lambda x} a \,b^{2}+2 \lambda \left (c +\frac {\lambda }{2}\right ) x}{\lambda }}d x}}-b \\ \end{align*}

✓ Mathematica. Time used: 1.079 (sec). Leaf size: 306

ode=D[y[x],x]==a*Exp[\[Lambda]*x]*y[x]^3+3*a*b*Exp[\[Lambda]*x]*y[x]^2+c*y[x]-2*a*b^3*Exp[\[Lambda]*x]+b*c; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -b-\frac {e^{c x-\frac {3 a b^2 e^{\lambda x}}{\lambda }}}{\sqrt {-2 \int _1^xa e^{(2 c+\lambda ) K[1]-\frac {6 a b^2 e^{\lambda K[1]}}{\lambda }}dK[1]+c_1}}\\ y(x)&\to -b+\frac {e^{c x-\frac {3 a b^2 e^{\lambda x}}{\lambda }}}{\sqrt {-2 \int _1^xa e^{(2 c+\lambda ) K[1]-\frac {6 a b^2 e^{\lambda K[1]}}{\lambda }}dK[1]+c_1}}\\ y(x)&\to -b\\ y(x)&\to \frac {e^{c x-\frac {3 a b^2 e^{\lambda x}}{\lambda }}}{\sqrt {2} \sqrt {-\int _1^xa e^{(2 c+\lambda ) K[1]-\frac {6 a b^2 e^{\lambda K[1]}}{\lambda }}dK[1]}}-b\\ y(x)&\to -\frac {e^{c x-\frac {3 a b^2 e^{\lambda x}}{\lambda }}}{\sqrt {2} \sqrt {-\int _1^xa e^{(2 c+\lambda ) K[1]-\frac {6 a b^2 e^{\lambda K[1]}}{\lambda }}dK[1]}}-b \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(2*a*b**3*exp(lambda_*x) - 3*a*b*y(x)**2*exp(lambda_*x) - a*y(x)**3*exp(lambda_*x) - b*c - c*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE 2*a*b**3*exp(lambda_*x) - 3*a*b*y(x)**2*exp(lambda_*x) - a*y(x)**3*exp(lambda_*x) - b*c - c*y(x) + Derivative(y(x), x) cannot be solved by the lie group method