25.5.1 problem 1

Internal problem ID [6879]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 6
Problem number : 1
Date solved : Tuesday, September 30, 2025 at 03:59:52 PM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }-a y+y^{2}&=x^{-2 a} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 74

ode:=x*diff(y(x),x)-a*y(x)+y(x)^2 = x^(-2*a); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-x^{-a} c_1 +a \right ) \sinh \left (\frac {x^{-a}}{a}\right )+\left (c_1 a -x^{-a}\right ) \cosh \left (\frac {x^{-a}}{a}\right )}{\cosh \left (\frac {x^{-a}}{a}\right ) c_1 +\sinh \left (\frac {x^{-a}}{a}\right )} \]

✓ Mathematica. Time used: 0.238 (sec). Leaf size: 112

ode=x*D[y[x],x]-a*y[x]+y[x]^2==x^(-2*a); 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*y(x) + x*Derivative(y(x), x) + y(x)**2 - 1/x**(2*a),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

RecursionError : maximum recursion depth exceeded