Internal problem ID [10071]
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. Equations of
the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 77.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y^{\prime } y-\left (2 \ln \relax (x )+a +1\right ) y-x \left (-\ln \relax (x )^{2}-a \ln \relax (x )+b \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 140
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)
\[ y \relax (x ) = \frac {x \left (-\tanh \left (\frac {\RootOf \left (-\tanh \left (\frac {\textit {\_Z} \sqrt {a^{2}+4 b}}{2}\right ) {\mathrm e}^{\textit {\_Z}} \sqrt {a^{2}+4 b}+2 \,{\mathrm e}^{\textit {\_Z}} \ln \relax (x )+{\mathrm e}^{\textit {\_Z}} a -2 \left (\int _{}^{-\frac {\tanh \left (\frac {\textit {\_Z} \sqrt {a^{2}+4 b}}{2}\right ) \sqrt {a^{2}+4 b}}{2}+\frac {a}{2}}{\mathrm e}^{-\frac {2 \arctanh \left (\frac {2 \textit {\_a} -a}{\sqrt {a^{2}+4 b}}\right )}{\sqrt {a^{2}+4 b}}}d \textit {\_a} \right )+2 c_{1}\right ) \sqrt {a^{2}+4 b}}{2}\right ) \sqrt {a^{2}+4 b}+2 \ln \relax (x )+a \right )}{2} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==(2*Log[x]+a+1)*y[x]+x*( -(Log[x])^2-a*Log[x]+b),y[x],x,IncludeSingularSolutions -> True]
Not solved