55.26.26 problem 26

Internal problem ID [13773]
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 : 26
Date solved : Thursday, October 02, 2025 at 08:06:32 AM
CAS classification : [_Abel]

\begin{align*} y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{3}+3 a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x} y^{2}-2 a \,b^{3} {\mathrm e}^{\left (\lambda +3 \mu \right ) x}-b \mu \,{\mathrm e}^{\mu x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 164

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

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

✓ Mathematica. Time used: 1.836 (sec). Leaf size: 186

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

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

✗ Sympy

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

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