2.24.42 Problem 67
Internal
problem
ID
[13606]
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
:
67
Date
solved
:
Friday, December 19, 2025 at 08:23:25 AM
CAS
classification
:
[[_Abel, `2nd type`, `class A`]]
\begin{align*}
y y^{\prime }&=\left (a \,{\mathrm e}^{x}+b \right ) y+c \,{\mathrm e}^{2 x}-{\mathrm e}^{x} a b -b^{2} \\
\end{align*}
Unknown ode type.
2.24.42.1 ✓ Maple. Time used: 0.006 (sec). Leaf size: 153
ode:=y(x)*diff(y(x),x) = (a*exp(x)+b)*y(x)+c*exp(2*x)-a*b*exp(x)-b^2;
dsolve(ode,y(x), singsol=all);
\[
{\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {\left (b -y\right ) a -2 \,{\mathrm e}^{x} c}{\sqrt {a^{2}+4 c}\, \left (b -y\right )}\right )}{\sqrt {a^{2}+4 c}}} \sqrt {\frac {c \,{\mathrm e}^{2 x}-\left (b -y\right ) \left (a \,{\mathrm e}^{x}+b -y\right )}{\left (b -y\right )^{2}}}\, y-\int _{}^{\frac {{\mathrm e}^{x}}{-b +y}}\frac {\sqrt {\textit {\_a}^{2} c +\textit {\_a} a -1}\, {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} c +a}{\sqrt {a^{2}+4 c}}\right )}{\sqrt {a^{2}+4 c}}}}{\textit {\_a}}d \textit {\_a} b +c_1 = 0
\]
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
Looking for potential symmetries
found: 2 potential symmetries. Proceeding with integration step
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=\left (a \,{\mathrm e}^{x}+b \right ) y \left (x \right )+c \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}-b^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\left (a \,{\mathrm e}^{x}+b \right ) y \left (x \right )+c \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}-b^{2}}{y \left (x \right )} \end {array} \]
2.24.42.2 ✗ Mathematica
ode=y[x]*D[y[x],x]==(a*Exp[x]+b)*y[x]+c*Exp[2*x]-a*b*Exp[x]-b^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.24.42.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(a*b*exp(x) + b**2 - c*exp(2*x) - (a*exp(x) + b)*y(x) + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out