77.1.7 problem 18 (page 30)

Internal problem ID [17818]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 18 (page 30)
Date solved : Thursday, March 13, 2025 at 10:56:37 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y^{\prime }&=\frac {2 x y}{x^{2}+y^{2}} \end{align*}

✓ Maple. Time used: 0.168 (sec). Leaf size: 47

ode:=diff(y(x),x) = 2*x*y(x)/(x^2+y(x)^2); 
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: 0.873 (sec). Leaf size: 70

ode=D[y[x],x]==(2*x*y[x])/(x^2+y[x]^2); 
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: 1.262 (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)
 

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