2.26.21 Problem 22

Internal problem ID [13657]
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 : 22
Date solved : Friday, December 19, 2025 at 10:08:10 AM
CAS classification : [_Abel]

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

Unknown ode type.

2.26.21.1 ✓ Maple. Time used: 0.001 (sec). Leaf size: 170

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

\[ \frac {2 \,{\mathrm e}^{\frac {-2 a^{2} \left (\lambda ^{2} x -\frac {9 y^{2}}{2}-3 \lambda \right ) {\mathrm e}^{2 \lambda x}-18 a^{3} y \,{\mathrm e}^{3 \lambda x}+9 \,{\mathrm e}^{4 \lambda x} a^{4}+4 \lambda \left (y a \left (\lambda x -\frac {3}{2}\right ) {\mathrm e}^{\lambda x}-\frac {x \lambda y^{2}}{2}+\frac {\lambda }{4}\right )}{2 \lambda \left ({\mathrm e}^{\lambda x} a -y\right )^{2}}} \sqrt {-\lambda }+3 a \left (\operatorname {erf}\left (\frac {\left (3 a^{2} {\mathrm e}^{2 \lambda x}-3 y \,{\mathrm e}^{\lambda x} a +\lambda \right ) \sqrt {2}}{\sqrt {-\lambda }\, \left (2 \,{\mathrm e}^{\lambda x} a -2 y\right )}\right ) \sqrt {2}\, \sqrt {\pi }+2 c_1 \right )}{6 a} = 0 \]

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
trying Abel 
<- Abel successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-y \left (x \right )^{3}+3 a^{2} {\mathrm e}^{2 \lambda x} y \left (x \right )-2 a^{3} {\mathrm e}^{3 \lambda x}+a \lambda \,{\mathrm e}^{\lambda x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-y \left (x \right )^{3}+3 a^{2} {\mathrm e}^{2 \lambda x} y \left (x \right )-2 a^{3} {\mathrm e}^{3 \lambda x}+a \lambda \,{\mathrm e}^{\lambda x} \end {array} \]

2.26.21.2 ✓ Mathematica. Time used: 0.965 (sec). Leaf size: 129

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

\[ \text {Solve}\left [-\frac {3 a e^{\lambda x}}{\sqrt {\lambda }}=\frac {2 \exp \left (\frac {1}{2} \left (-\frac {3 a e^{\lambda x}}{\sqrt {\lambda }}-\frac {1}{\frac {a e^{\lambda x}}{\sqrt {\lambda }}-\frac {y(x)}{\sqrt {\lambda }}}\right )^2\right )}{\sqrt {2 \pi } \text {erfi}\left (\frac {-\frac {3 a e^{\lambda x}}{\sqrt {\lambda }}-\frac {1}{\frac {a e^{\lambda x}}{\sqrt {\lambda }}-\frac {y(x)}{\sqrt {\lambda }}}}{\sqrt {2}}\right )+2 c_1},y(x)\right ] \]

2.26.21.3 ✗ Sympy

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

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