2.24.49 Problem 77
Internal
problem
ID
[13613]
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
:
77
Date
solved
:
Friday, December 19, 2025 at 08:41:56 AM
CAS
classification
:
[[_Abel, `2nd type`, `class A`]]
\begin{align*}
y y^{\prime }&=\left (2 \ln \left (x \right )+a +1\right ) y+x \left (-\ln \left (x \right )^{2}-a \ln \left (x \right )+b \right ) \\
\end{align*}
Unknown ode type.
2.24.49.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 163
ode:=y(x)*diff(y(x),x) = (2*ln(x)+a+1)*y(x)+x*(-ln(x)^2-a*ln(x)+b);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {x \left (-\tanh \left (\frac {\operatorname {RootOf}\left ({\mathrm e}^{-\frac {2 \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} -a}{\sqrt {a^{2}+4 b}}\right )}{\sqrt {a^{2}+4 b}}} \tanh \left (\frac {\textit {\_Z} \sqrt {a^{2}+4 b}}{2}\right ) \sqrt {a^{2}+4 b}-\sqrt {a^{2}+4 b}\, \tanh \left (\frac {\textit {\_Z} \sqrt {a^{2}+4 b}}{2}\right ) {\mathrm e}^{\textit {\_Z}}+2 \ln \left (x \right ) {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{-\frac {2 \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} -a}{\sqrt {a^{2}+4 b}}\right )}{\sqrt {a^{2}+4 b}}} a +{\mathrm e}^{\textit {\_Z}} a +2 c_1 \right ) \sqrt {a^{2}+4 b}}{2}\right ) \sqrt {a^{2}+4 b}+2 \ln \left (x \right )+a \right )}{2}
\]
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
found: 2 potential symmetries. Proceeding with integration step
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=\left (2 \ln \left (x \right )+a +1\right ) y \left (x \right )+x \left (-\ln \left (x \right )^{2}-a \ln \left (x \right )+b \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\left (2 \ln \left (x \right )+a +1\right ) y \left (x \right )+x \left (-\ln \left (x \right )^{2}-a \ln \left (x \right )+b \right )}{y \left (x \right )} \end {array} \]
2.24.49.2 ✗ Mathematica
ode=y[x]*D[y[x],x]==(2*Log[x]+a+1)*y[x]+x*( -(Log[x])^2-a*Log[x]+b);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.24.49.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-x*(-a*log(x) + b - log(x)**2) - (a + 2*log(x) + 1)*y(x) + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out
Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0]
Sympy version 1.14.0