73.8.45 problem 13.7 (e)

Internal problem ID [15253]
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 : Tuesday, January 28, 2025 at 07:50:58 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.946 (sec). Leaf size: 11

dsolve([diff(y(x),x$2)=-diff(y(x),x)*exp(-y(x)),y(0) = 0, D(y)(0) = 2],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 5.474 (sec). Leaf size: 13

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

\[ y(x)\to \log \left (2 e^x-1\right ) \]