31.5.2 problem 2

Internal problem ID [5745]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 6
Problem number : 2
Date solved : Monday, January 27, 2025 at 01:12:27 PM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }-a y+y^{2}&=x^{-\frac {2 a}{3}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 119

dsolve(x*diff(y(x),x)-a*y(x)+y(x)^2=x^(-2*a/3),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\left (\left (3 x^{-\frac {a}{3}}-2 a \right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a -x^{-\frac {a}{3}}\right )\right ) {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+c_{1} \left (\left (2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a +x^{-\frac {a}{3}}\right )\right )}{\left (-3 \sqrt {x^{-\frac {2 a}{3}}}+a \right ) {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+c_{1} \left (3 \sqrt {x^{-\frac {2 a}{3}}}+a \right )} \]

✓ Solution by Mathematica

Time used: 0.418 (sec). Leaf size: 270

DSolve[x*D[y[x],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^2 c_1 x^{2 a/3}+3 i a x^{a/3}+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 {\left (a^2 x^{2 a/3}+3\right ) \sinh \left (\frac {3 x^{-a/3}}{a}\right )-3 a x^{a/3} \cosh \left (\frac {3 x^{-a/3}}{a}\right )}{a x^{2 a/3} \sinh \left (\frac {3 x^{-a/3}}{a}\right )-3 x^{a/3} \cosh \left (\frac {3 x^{-a/3}}{a}\right )} \\ \end{align*}