61.24.77 problem 77

Internal problem ID [12411]
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 : Monday, March 31, 2025 at 05:32:26 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*}

✓ Maple. Time used: 0.003 (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);
 

\[ y = \frac {x \left (-\tanh \left (\frac {\operatorname {RootOf}\left (-\tanh \left (\frac {\textit {\_Z} \sqrt {a^{2}+4 b}}{2}\right ) {\mathrm e}^{\textit {\_Z}} \sqrt {a^{2}+4 b}+{\mathrm e}^{-\frac {2 \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} -a}{\sqrt {a^{2}+4 b}}\right )}{\sqrt {a^{2}+4 b}}} \tanh \left (\frac {\textit {\_Z} \sqrt {a^{2}+4 b}}{2}\right ) \sqrt {a^{2}+4 b}+2 \,{\mathrm e}^{\textit {\_Z}} \ln \left (x \right )+{\mathrm e}^{\textit {\_Z}} a -{\mathrm e}^{-\frac {2 \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} -a}{\sqrt {a^{2}+4 b}}\right )}{\sqrt {a^{2}+4 b}}} a +2 c_1 \right ) \sqrt {a^{2}+4 b}}{2}\right ) \sqrt {a^{2}+4 b}+2 \ln \left (x \right )+a \right )}{2} \]

✗ 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

✗ 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