53.1.8 problem 8

Internal problem ID [8442]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 94. Factoring the left member. EXERCISES Page 309
Problem number : 8
Date solved : Sunday, March 30, 2025 at 01:04:59 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (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: 0.046 (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: 0.493 (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)
 

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