23.2.13 problem 13

Internal problem ID [5368]
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 : 13
Date solved : Tuesday, September 30, 2025 at 12:36:17 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=diff(y(x),x)^2 = x^2*y(x)^2; 
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: 0.027 (sec). Leaf size: 38

ode=(D[y[x],x])^2==x^2*y[x]^2; 
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: 0.251 (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)
 

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