89.2.2 problem 2

Internal problem ID [24267]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 27
Problem number : 2
Date solved : Thursday, October 02, 2025 at 10:02:35 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 35

ode:=x*y(x)-(x^2+2*y(x)^2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 5.469 (sec). Leaf size: 65

ode=x*y[x]-(x^2+2*y[x]^2)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {x}{\sqrt {2} \sqrt {W\left (\frac {1}{2} e^{-c_1} x^2\right )}}\\ y(x)&\to \frac {x}{\sqrt {2} \sqrt {W\left (\frac {1}{2} e^{-c_1} x^2\right )}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.746 (sec). Leaf size: 19

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

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