2.24.47 Problem 73
Internal
problem
ID
[13611]
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
:
73
Date
solved
:
Friday, December 19, 2025 at 08:38:19 AM
CAS
classification
:
[[_Abel, `2nd type`, `class A`]]
\begin{align*}
y y^{\prime }-a \left (1+2 n +2 n \left (n +1\right ) x \right ) {\mathrm e}^{\left (n +1\right ) x} y&=-a^{2} n \left (n +1\right ) \left (n x +1\right ) x \,{\mathrm e}^{2 \left (n +1\right ) x} \\
\end{align*}
Unknown ode type.
2.24.47.1 ✓ Maple. Time used: 0.006 (sec). Leaf size: 130
ode:=y(x)*diff(y(x),x)-a*(1+2*n+2*n*(n+1)*x)*exp((n+1)*x)*y(x) = -a^2*n*(n+1)*(n*x+1)*x*exp(2*(n+1)*x);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {a \left (1+2 n^{2} x +\left (\tan \left (\frac {\operatorname {RootOf}\left (2 x \,n^{2} {\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, \tan \left (\frac {\textit {\_a} \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}{2}\right ) \textit {\_Z} n +2 n x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+n \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 c_1 n \,{\mathrm e}^{\textit {\_a}}+{\mathrm e}^{\textit {\_Z} +\textit {\_a}}\right ) \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}{2}\right ) \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}+2 x +1\right ) n \right ) {\mathrm e}^{\left (n +1\right ) x}}{2 n +2}
\]
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
Looking for potential symmetries
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-a \left (27223+370545864 x \right ) {\mathrm e}^{13612 x} y \left (x \right )=-185272932 a^{2} \left (13611 x +1\right ) x \,{\mathrm e}^{27224 x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {a \left (27223+370545864 x \right ) {\mathrm e}^{13612 x} y \left (x \right )-185272932 a^{2} \left (13611 x +1\right ) x \,{\mathrm e}^{27224 x}}{y \left (x \right )} \end {array} \]
2.24.47.2 ✗ Mathematica
ode=y[x]*D[y[x],x]-a*(1+2*n+2*n*(n+1)*x)*Exp[(n+1)*x]*y[x]==-a^2*n*(n+1)*(1+n*x)*x*Exp[2*(n+1)*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.24.47.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
n = symbols("n")
y = Function("y")
ode = Eq(a**2*n*x*(n + 1)*(n*x + 1)*exp(x*(2*n + 2)) - a*(2*n*x*(n + 1) + 2*n + 1)*y(x)*exp(x*(n + 1)) + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out