61.24.77 problem 77

Internal problem ID [12490]
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 : Tuesday, January 28, 2025 at 08:01:51 PM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} y^{\prime } y&=\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*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 163

dsolve(y(x)*diff(y(x),x)=(2*ln(x)+a+1)*y(x)+x*( -(ln(x))^2-a*ln(x)+b),y(x), singsol=all)
 

\[ y = \frac {x \left (-\tanh \left (\frac {\operatorname {RootOf}\left (-\tanh \left (\frac {\textit {\_Z} \sqrt {a^{2}+4 b}}{2}\right ) {\mathrm e}^{\textit {\_Z}} \sqrt {a^{2}+4 b}+{\mathrm e}^{-\frac {2 \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} -a}{\sqrt {a^{2}+4 b}}\right )}{\sqrt {a^{2}+4 b}}} \tanh \left (\frac {\textit {\_Z} \sqrt {a^{2}+4 b}}{2}\right ) \sqrt {a^{2}+4 b}+2 \,{\mathrm e}^{\textit {\_Z}} \ln \left (x \right )+a \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{-\frac {2 \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} -a}{\sqrt {a^{2}+4 b}}\right )}{\sqrt {a^{2}+4 b}}} a +2 c_{1} \right ) \sqrt {a^{2}+4 b}}{2}\right ) \sqrt {a^{2}+4 b}+2 \ln \left (x \right )+a \right )}{2} \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[y[x]*D[y[x],x]==(2*Log[x]+a+1)*y[x]+x*( -(Log[x])^2-a*Log[x]+b),y[x],x,IncludeSingularSolutions -> True]
 

Not solved