67.8.45 problem 13.7 (e)

Internal problem ID [16540]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.7 (e)
Date solved : Thursday, October 02, 2025 at 01:36:14 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

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

With initial conditions

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

✓ Maple. Time used: 0.258 (sec). Leaf size: 11

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

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

✓ Mathematica. Time used: 5.015 (sec). Leaf size: 13

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

\begin{align*} y(x)&\to \log \left (2 e^x-1\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE exp(y(x))*Derivative(y(x), (x, 2)) + Derivative(y(x), x) cannot be solved by the factorable group method