Internal problem ID [3893]
Book: Differential Equations, By George Boole F.R.S. 1865
Section: Chapter 6
Problem number: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x -a y+y^{2}-x^{-\frac {2 a}{3}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 163
dsolve(x*diff(y(x),x)-a*y(x)+y(x)^2=x^(-2*a/3),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (3 x^{-\frac {a}{3}} \sqrt {x^{-\frac {2 a}{3}}}-x^{-\frac {a}{3}} a -2 \sqrt {x^{-\frac {2 a}{3}}}\, a +a^{2}\right ) {\mathrm e}^{\frac {3 x^{-\frac {a}{3}}}{a}}+\left (-3 x^{-\frac {a}{3}} \sqrt {x^{-\frac {2 a}{3}}}\, c_{1}-x^{-\frac {a}{3}} c_{1} a -2 \sqrt {x^{-\frac {2 a}{3}}}\, c_{1} a -c_{1} a^{2}\right ) {\mathrm e}^{-\frac {3 x^{-\frac {a}{3}}}{a}}}{\left (3 \sqrt {x^{-\frac {2 a}{3}}}-a \right ) {\mathrm e}^{\frac {3 x^{-\frac {a}{3}}}{a}}+\left (3 \sqrt {x^{-\frac {2 a}{3}}}\, c_{1}+c_{1} a \right ) {\mathrm e}^{-\frac {3 x^{-\frac {a}{3}}}{a}}} \]
✓ Solution by Mathematica
Time used: 0.441 (sec). Leaf size: 208
DSolve[x*y'[x]-a*y[x]+y[x]^2==x^(-2*a/3),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-a/3} \left (\left (a^2 x^{2 a/3}-3 i a c_1 x^{a/3}+3\right ) \cosh \left (\frac {3 x^{-a/3}}{a}\right )+i \left (a x^{a/3} \left (a c_1 x^{a/3}+3 i\right )+3 c_1\right ) \sinh \left (\frac {3 x^{-a/3}}{a}\right )\right )}{\left (a x^{a/3}-3 i c_1\right ) \cosh \left (\frac {3 x^{-a/3}}{a}\right )+i \left (a c_1 x^{a/3}+3 i\right ) \sinh \left (\frac {3 x^{-a/3}}{a}\right )} \\ y(x)\to \frac {3}{a x^{2 a/3}-3 x^{a/3} \coth \left (\frac {3 x^{-a/3}}{a}\right )}+a \\ \end{align*}