23.2.211 problem 217

Internal problem ID [5566]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 217
Date solved : Tuesday, September 30, 2025 at 12:53:23 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 31

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

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

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 50

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

\begin{align*} y(x)&\to \frac {c_1}{x}\\ y(x)&\to -\sqrt {x^2+2 c_1}\\ y(x)&\to \sqrt {x^2+2 c_1}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.340 (sec). Leaf size: 27

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

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