50.4.18 problem 18

Internal problem ID [7867]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.5. Exact Equations. Page 20
Problem number : 18
Date solved : Sunday, March 30, 2025 at 12:33:33 PM
CAS classification : [_separable]

\begin{align*} \frac {x}{\left (x^{2}+y^{2}\right )^{{3}/{2}}}+\frac {y y^{\prime }}{\left (x^{2}+y^{2}\right )^{{3}/{2}}}&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 27

ode:=x/(x^2+y(x)^2)^(3/2)+y(x)/(x^2+y(x)^2)^(3/2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.095 (sec). Leaf size: 39

ode=(x/(x^2+y[x]^2)^(3/2))+(y[x]/(x^2+y[x]^2)^(3/2))*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\sqrt {-x^2+2 c_1} \\ y(x)\to \sqrt {-x^2+2 c_1} \\ \end{align*}

✓ Sympy. Time used: 0.334 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x/(x**2 + y(x)**2)**(3/2) + y(x)*Derivative(y(x), x)/(x**2 + y(x)**2)**(3/2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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