89.28.7 problem 7

Internal problem ID [24895]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Exercises at page 229
Problem number : 7
Date solved : Thursday, October 02, 2025 at 10:49:10 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 23

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

\begin{align*} y &= c_1 \,{\mathrm e}^{\frac {x^{2}}{2}} \\ y &= c_1 \,{\mathrm e}^{-\frac {x^{2}}{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 38

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

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

✓ Sympy. Time used: 0.254 (sec). Leaf size: 20

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

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