29.20.4 problem 549

Internal problem ID [5145]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 20
Problem number : 549
Date solved : Monday, January 27, 2025 at 10:15:39 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} x \left (x -2 y\right ) y^{\prime }+\left (2 x -y\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 69

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

\begin{align*} y \left (x \right ) &= \frac {c_{1}^{2} x^{2}+\sqrt {c_{1} x \left (c_{1}^{3} x^{3}+4\right )}}{2 c_{1}^{2} x} \\ y \left (x \right ) &= \frac {c_{1}^{2} x^{2}-\sqrt {c_{1} x \left (c_{1}^{3} x^{3}+4\right )}}{2 c_{1}^{2} x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.688 (sec). Leaf size: 114

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

\begin{align*} y(x)\to \frac {1}{2} \left (x-\frac {\sqrt {x^3-4 e^{c_1}}}{\sqrt {x}}\right ) \\ y(x)\to \frac {1}{2} \left (x+\frac {\sqrt {x^3-4 e^{c_1}}}{\sqrt {x}}\right ) \\ y(x)\to \frac {x}{2}-\frac {\sqrt {x^3}}{2 \sqrt {x}} \\ y(x)\to \frac {x^{3/2}+\sqrt {x^3}}{2 \sqrt {x}} \\ \end{align*}