89.1.29 problem 29

Internal problem ID [24264]
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 : 29
Date solved : Thursday, October 02, 2025 at 10:02:29 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 19

ode:=diff(y(x),x) = x*exp(-y(x)-x^2); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \ln \left (-\frac {1}{2}+\frac {3 \,{\mathrm e}^{x^{2}}}{2}\right )-x^{2} \]

✓ Mathematica. Time used: 1.853 (sec). Leaf size: 21

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

\begin{align*} y(x)&\to \log \left (\frac {3}{2}-\frac {e^{-x^2}}{2}\right ) \end{align*}

✓ Sympy. Time used: 0.145 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = \log {\left (\frac {3}{2} - \frac {e^{- x^{2}}}{2} \right )} \]