82.12.28 problem Ex. 30

Internal problem ID [18714]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Examples on chapter II at page 29
Problem number : Ex. 30
Date solved : Thursday, March 13, 2025 at 12:40:45 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 47

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

\begin{align*} y \left (x \right ) &= \frac {1-\sqrt {-4 x^{2} c_{1}^{2}+1}}{2 c_{1}} \\ y \left (x \right ) &= \frac {1+\sqrt {-4 x^{2} c_{1}^{2}+1}}{2 c_{1}} \\ \end{align*}

✓ Mathematica. Time used: 0.984 (sec). Leaf size: 66

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

\begin{align*} y(x)\to \frac {1}{2} \left (e^{c_1}-\sqrt {-4 x^2+e^{2 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: 1.592 (sec). Leaf size: 44

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)
 

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