23.1.642 problem 636

Internal problem ID [5249]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 636
Date solved : Tuesday, September 30, 2025 at 12:00:14 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 23

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

\[ y = \sqrt {\frac {1}{a \operatorname {LambertW}\left (\frac {c_1 \,x^{2}}{a}\right )}}\, x \]

✓ Mathematica. Time used: 12.112 (sec). Leaf size: 71

ode=(x^2+a*y[x]^2)*D[y[x],x]==x*y[x]; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 0.736 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-x*y(x) + (a*y(x)**2 + x**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = e^{C_{1} + \frac {W\left (\frac {x^{2} e^{- 2 C_{1}}}{a}\right )}{2}} \]