55.24.72 problem 72

Internal problem ID [13701]
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 1.3.3-2.
Problem number : 72
Date solved : Sunday, October 12, 2025 at 04:42:43 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 53

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

\[ y = -\frac {\left (x \operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} b x -\operatorname {Ei}_{1}\left (-\textit {\_Z} \right )+c_1 \right ) b -1\right ) a \,{\mathrm e}^{b x}}{b \operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} b x -\operatorname {Ei}_{1}\left (-\textit {\_Z} \right )+c_1 \right )} \]

✓ Mathematica. Time used: 0.283 (sec). Leaf size: 59

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

\[ \text {Solve}\left [b x e^{\frac {a e^{b x}}{a b x e^{b x}+b y(x)}}=\operatorname {ExpIntegralEi}\left (\frac {a e^{b x}}{a b e^{b x} x+b y(x)}\right )+c_1,y(x)\right ] \]

✗ Sympy

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

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