Internal problem ID [4400]
Book: Differential Equations, By George Boole F.R.S. 1865
Section: Chapter 6
Problem number: 1.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {x y^{\prime }-a y+y^{2}=x^{-2 a}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 74
dsolve(x*diff(y(x),x)-a*y(x)+y(x)^2=x^(-2*a),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-x^{-a} c_{1} +a \right ) \sinh \left (\frac {x^{-a}}{a}\right )+\left (c_{1} a -x^{-a}\right ) \cosh \left (\frac {x^{-a}}{a}\right )}{\cosh \left (\frac {x^{-a}}{a}\right ) c_{1} +\sinh \left (\frac {x^{-a}}{a}\right )} \]
✓ Solution by Mathematica
Time used: 0.393 (sec). Leaf size: 112
DSolve[x*y'[x]-a*y[x]+y[x]^2==x^(-2*a),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-a} \left (\left (a x^a+i c_1\right ) \cosh \left (\frac {x^{-a}}{a}\right )-i \left (a c_1 x^a-i\right ) \sinh \left (\frac {x^{-a}}{a}\right )\right )}{\cosh \left (\frac {x^{-a}}{a}\right )-i c_1 \sinh \left (\frac {x^{-a}}{a}\right )} y(x)\to a-x^{-a} \coth \left (\frac {x^{-a}}{a}\right ) \end{align*}