89.33.18 problem 19

Internal problem ID [25002]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 17. Special Equations of order Two. Exercises at page 251
Problem number : 19
Date solved : Thursday, October 02, 2025 at 11:46:12 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

With initial conditions

\begin{align*} y \left (3\right )&=0 \\ y^{\prime }\left (3\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 12

ode:=diff(diff(y(x),x),x)+exp(-2*y(x)) = 0; 
ic:=[y(3) = 0, D(y)(3) = -1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\ln \left (\left (x -4\right )^{2}\right )}{2} \]

✓ Mathematica. Time used: 0.085 (sec). Leaf size: 11

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

\begin{align*} y(x)&\to \log (4-x) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + exp(-2*y(x)),0) 
ics = {y(3): 0, Subs(Derivative(y(x), x), x, 3): -1} 
dsolve(ode,func=y(x),ics=ics)