2.24.46 Problem 72
Internal
problem
ID
[13610]
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
:
Friday, December 19, 2025 at 08:37:18 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*}
Unknown ode type.
2.24.46.1 ✓ Maple. Time used: 0.002 (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);
\[
y = -\frac {\left (x \operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} b x -\operatorname {Ei}_{1}\left (-\textit {\_Z} \right )+c_1 \right ) b -1\right ) a \,{\mathrm e}^{b x}}{b \operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} b x -\operatorname {Ei}_{1}\left (-\textit {\_Z} \right )+c_1 \right )}
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+a \left (2 b x +1\right ) {\mathrm e}^{b x} y \left (x \right )=-a^{2} b \,x^{2} {\mathrm e}^{2 b x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-\frac {a \left (2 b x +1\right ) {\mathrm e}^{b x} y \left (x \right )+a^{2} b \,x^{2} {\mathrm e}^{2 b x}}{y \left (x \right )} \end {array} \]
2.24.46.2 ✓ 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]
\[
\text {Solve}\left [b x e^{\frac {a e^{b x}}{a b x e^{b x}+b y(x)}}=\operatorname {ExpIntegralEi}\left (\frac {a e^{b x}}{a b e^{b x} x+b y(x)}\right )+c_1,y(x)\right ]
\]
2.24.46.3 ✗ 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