76.16.1 problem Ex. 1

Internal problem ID [20100]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Problems at page 36
Problem number : Ex. 1
Date solved : Thursday, October 02, 2025 at 05:25:45 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 33

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

\begin{align*} y &= 0 \\ y &= c_1 \,x^{\frac {1}{2}+\frac {\sqrt {5}}{2}} \\ y &= c_1 \,x^{-\frac {\sqrt {5}}{2}+\frac {1}{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 48

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

\begin{align*} y(x)&\to c_1 x^{\frac {1}{2}-\frac {\sqrt {5}}{2}}\\ y(x)&\to c_1 x^{\frac {1}{2} \left (1+\sqrt {5}\right )}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.141 (sec). Leaf size: 31

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

\[ \left [ y{\left (x \right )} = \frac {C_{1}}{x^{- \frac {1}{2} + \frac {\sqrt {5}}{2}}}, \ y{\left (x \right )} = C_{1} x^{\frac {1}{2} + \frac {\sqrt {5}}{2}}\right ] \]