61.23.6 problem 6

Internal problem ID [12407]
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.
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 07:58:14 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime } y&=\left (\frac {a}{x^{{2}/{3}}}-\frac {2}{3 a \,x^{{1}/{3}}}\right ) y+1 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 187

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)
 

\[ \frac {-\sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}\, \operatorname {BesselI}\left (1, \frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right ) c_{1} a^{2}+\operatorname {BesselK}\left (1, -\frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right ) \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}\, a^{2}+x^{{1}/{3}} \operatorname {BesselI}\left (0, \frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right ) c_{1} -x^{{1}/{3}} \operatorname {BesselK}\left (0, -\frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right )}{-\operatorname {BesselI}\left (1, \frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right ) \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}\, a^{2}+x^{{1}/{3}} \operatorname {BesselI}\left (0, \frac {2 \sqrt {\frac {x^{{2}/{3}}+a y}{a^{4}}}}{3}\right )} = 0 \]

✗ Solution by Mathematica

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

DSolve[y[x]*D[y[x],x]==(a*x^(-2/3)-2/3*a^(-1)*x^(-1/3))*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
 

Not solved