31.5.1 problem 1

Internal problem ID [5744]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 6
Problem number : 1
Date solved : Monday, January 27, 2025 at 01:12:26 PM
CAS classification : [_rational, _Riccati]

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

✓ Solution by Maple

Time used: 0.003 (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: 0.365 (sec). Leaf size: 112

DSolve[x*D[y[x],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*}