89.1.3 problem 3

Internal problem ID [24238]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 21
Problem number : 3
Date solved : Thursday, October 02, 2025 at 10:01:13 PM
CAS classification : [_separable]

\begin{align*} x y^{3}+{\mathrm e}^{x^{2}} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

ode:=x*y(x)^3+exp(x^2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {1}{\sqrt {c_1 -{\mathrm e}^{-x^{2}}}} \\ y &= -\frac {1}{\sqrt {c_1 -{\mathrm e}^{-x^{2}}}} \\ \end{align*}

✓ Mathematica. Time used: 7.096 (sec). Leaf size: 70

ode=(x*y[x]^3)+(Exp[x^2])*D[y[x],{x,1}]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.519 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x)**3 + exp(x**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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