79.2.3 problem (d)

Internal problem ID [21085]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 1. First order equations: Some integrable cases. Excercises XIII at page 24
Problem number : (d)
Date solved : Thursday, October 02, 2025 at 07:07:00 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {{\mathrm e}^{-y^{2}}}{y \left (x^{2}+2 x \right )} \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.218 (sec). Leaf size: 43

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

\begin{align*} y &= \sqrt {\ln \left (1+\ln \left (2\right )-\ln \left (\frac {x +2}{x}\right )\right )} \\ y &= -\sqrt {\ln \left (1+\ln \left (2\right )-\ln \left (\frac {x +2}{x}\right )\right )} \\ \end{align*}

✓ Mathematica. Time used: 1.563 (sec). Leaf size: 45

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

\begin{align*} y(x)&\to -\sqrt {\log (\log (2 x)-\log (x+2)+1)}\\ y(x)&\to \sqrt {\log (\log (2 x)-\log (x+2)+1)} \end{align*}

✓ Sympy. Time used: 0.759 (sec). Leaf size: 42

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

\[ \left [ y{\left (x \right )} = - \sqrt {\log {\left (\log {\left (x \right )} - \log {\left (x + 2 \right )} + \log {\left (2 \right )} + 1 \right )}}, \ y{\left (x \right )} = \sqrt {\log {\left (\log {\left (x \right )} - \log {\left (x + 2 \right )} + \log {\left (2 \right )} + 1 \right )}}\right ] \]