Internal problem ID [9988]
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 1.3.2. Equations of the
form \(y y'=f(x) y+1\)
Problem number: 6.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y^{\prime } y-\left (\frac {a}{x^{\frac {2}{3}}}-\frac {2}{3 a \,x^{\frac {1}{3}}}\right ) y-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 128
dsolve(y(x)*diff(y(x),x)=(a*x^(-2/3)-2/3*a^(-1)*x^(-1/3))*y(x)+1,y(x), singsol=all)
\[ c_{1}+\frac {\BesselK \left (1, -\frac {2 \sqrt {\frac {x^{\frac {2}{3}}+a y \relax (x )}{a^{4}}}}{3}\right ) \sqrt {\frac {x^{\frac {2}{3}}+a y \relax (x )}{a^{4}}}\, a^{2}-x^{\frac {1}{3}} \BesselK \left (0, -\frac {2 \sqrt {\frac {x^{\frac {2}{3}}+a y \relax (x )}{a^{4}}}}{3}\right )}{-\BesselI \left (1, \frac {2 \sqrt {\frac {x^{\frac {2}{3}}+a y \relax (x )}{a^{4}}}}{3}\right ) \sqrt {\frac {x^{\frac {2}{3}}+a y \relax (x )}{a^{4}}}\, a^{2}+x^{\frac {1}{3}} \BesselI \left (0, \frac {2 \sqrt {\frac {x^{\frac {2}{3}}+a y \relax (x )}{a^{4}}}}{3}\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==(a*x^(-2/3)-2/3*a^(-1)*x^(-1/3))*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
Not solved