31.3.2 problem 4

Internal problem ID [5732]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 4
Problem number : 4
Date solved : Monday, January 27, 2025 at 01:11:50 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} \frac {1}{x}+\frac {y^{\prime }}{y}+\frac {2}{y}-\frac {2 y^{\prime }}{x}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 51

dsolve(1/x+1/y(x)*diff(y(x),x)+2*(1/y(x)-1/x*diff(y(x),x))=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {c_{1} x -\sqrt {5 c_{1}^{2} x^{2}+4}}{2 c_{1}} \\ y \left (x \right ) &= \frac {c_{1} x +\sqrt {5 c_{1}^{2} x^{2}+4}}{2 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.466 (sec). Leaf size: 102

DSolve[1/x+1/y[x]*D[y[x],x]+2*(1/y[x]-1/x*D[y[x],x])==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{2} \left (x-\sqrt {5 x^2-4 e^{c_1}}\right ) \\ y(x)\to \frac {1}{2} \left (x+\sqrt {5 x^2-4 e^{c_1}}\right ) \\ y(x)\to \frac {1}{2} \left (x-\sqrt {5} \sqrt {x^2}\right ) \\ y(x)\to \frac {1}{2} \left (\sqrt {5} \sqrt {x^2}+x\right ) \\ \end{align*}