Internal problem ID [9924]
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. Form \(y y'-y=f(x)\). subsection 1.3.1-2. Solvable
equations and their solutions
Problem number: 18.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y^{\prime } y-y-\frac {2 a^{2}}{\sqrt {8 a^{2}+x^{2}}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 720
dsolve(y(x)*diff(y(x),x)-y(x)=2*a^2/sqrt(x^2+8*a^2),y(x), singsol=all)
\[ c_{1}+\frac {512 a \left (-\frac {33 x \left (a^{4}+\frac {23}{66} a^{2} x^{2}+\frac {1}{66} x^{4}\right ) \sqrt {8 a^{2}+x^{2}}}{64}+a^{6}+\frac {75 a^{4} x^{2}}{64}+\frac {27 a^{2} x^{4}}{128}+\frac {x^{6}}{128}\right ) {\mathrm e}^{-\frac {\left (x -y \relax (x )\right )^{2} \left (-64 \sqrt {8 a^{2}+x^{2}}\, a^{6}-108 \sqrt {8 a^{2}+x^{2}}\, a^{4} x^{2}-25 \sqrt {8 a^{2}+x^{2}}\, a^{2} x^{4}-\sqrt {8 a^{2}+x^{2}}\, x^{6}+328 a^{6} x +200 a^{4} x^{3}+29 a^{2} x^{5}+x^{7}\right )^{2}}{2 \left (128 a^{6}+150 a^{4} x^{2}-66 \sqrt {8 a^{2}+x^{2}}\, a^{4} x +27 a^{2} x^{4}-23 \sqrt {8 a^{2}+x^{2}}\, a^{2} x^{3}+x^{6}-\sqrt {8 a^{2}+x^{2}}\, x^{5}\right )^{2} a^{2} \left (-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}\right )}}+128 \sqrt {\pi }\, \erf \left (\frac {328 \left (x -y \relax (x )\right ) \left (\left (-\frac {8}{41} a^{6}-\frac {27}{82} a^{4} x^{2}-\frac {25}{328} a^{2} x^{4}-\frac {1}{328} x^{6}\right ) \sqrt {8 a^{2}+x^{2}}+x \left (a^{6}+\frac {25}{41} a^{4} x^{2}+\frac {29}{328} a^{2} x^{4}+\frac {1}{328} x^{6}\right )\right ) \sqrt {2}}{\sqrt {-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}}\, \left (\left (-132 x \,a^{5}-46 a^{3} x^{3}-2 x^{5} a \right ) \sqrt {8 a^{2}+x^{2}}+256 a^{7}+300 a^{5} x^{2}+54 a^{3} x^{4}+2 a \,x^{6}\right )}\right ) \sqrt {2}\, \sqrt {-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}}\, \left (\left (-\frac {33 a^{4} x}{64}+\frac {y \relax (x ) a^{4}}{4}-\frac {23 a^{2} x^{3}}{128}+\frac {21 a^{2} x^{2} y \relax (x )}{128}-\frac {x^{5}}{128}+\frac {x^{4} y \relax (x )}{128}\right ) \sqrt {8 a^{2}+x^{2}}+a^{6}+\frac {75 a^{4} x^{2}}{64}-\frac {25 a^{4} x y \relax (x )}{32}+\frac {27 a^{2} x^{4}}{128}-\frac {25 a^{2} x^{3} y \relax (x )}{128}+\frac {x^{6}}{128}-\frac {x^{5} y \relax (x )}{128}\right )}{\sqrt {-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}}\, \left (\left (-66 a^{4} x +32 y \relax (x ) a^{4}-23 a^{2} x^{3}+21 a^{2} x^{2} y \relax (x )-x^{5}+x^{4} y \relax (x )\right ) \sqrt {8 a^{2}+x^{2}}+128 a^{6}+150 a^{4} x^{2}-100 a^{4} x y \relax (x )+27 a^{2} x^{4}-25 a^{2} x^{3} y \relax (x )+x^{6}-x^{5} y \relax (x )\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==2*a^2/Sqrt[x^2+8*a^2],y[x],x,IncludeSingularSolutions -> True]
Not solved