29.23.3 problem 633

Internal problem ID [5225]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 633
Date solved : Monday, January 27, 2025 at 10:23:49 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 23

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

\[ y \left (x \right ) = \sqrt {\frac {1}{a \operatorname {LambertW}\left (\frac {c_{1} x^{2}}{a}\right )}}\, x \]

✓ Solution by Mathematica

Time used: 14.179 (sec). Leaf size: 71

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

\begin{align*} y(x)\to -\frac {x}{\sqrt {a} \sqrt {W\left (\frac {x^2 e^{-\frac {2 c_1}{a}}}{a}\right )}} \\ y(x)\to \frac {x}{\sqrt {a} \sqrt {W\left (\frac {x^2 e^{-\frac {2 c_1}{a}}}{a}\right )}} \\ y(x)\to 0 \\ \end{align*}