47.2.5 problem 5

Internal problem ID [7421]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 5
Date solved : Sunday, March 30, 2025 at 12:00:26 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.116 (sec). Leaf size: 47

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

\begin{align*} y &= \frac {1-\sqrt {4 c_1^{2} x^{2}+1}}{2 c_1} \\ y &= \frac {1+\sqrt {4 c_1^{2} x^{2}+1}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 1.187 (sec). Leaf size: 70

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

\begin{align*} y(x)\to \frac {1}{2} \left (-\sqrt {4 x^2+e^{2 c_1}}-e^{c_1}\right ) \\ y(x)\to \frac {1}{2} \left (\sqrt {4 x^2+e^{2 c_1}}-e^{c_1}\right ) \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.604 (sec). Leaf size: 46

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {4 x^{2} + e^{2 C_{1}}}}{2} - \frac {e^{C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {4 x^{2} + e^{2 C_{1}}}}{2} - \frac {e^{C_{1}}}{2}\right ] \]