56.5.22 problem 22

Internal problem ID [8983]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 22
Date solved : Monday, January 27, 2025 at 05:26:14 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Solution by Maple

Time used: 0.054 (sec). Leaf size: 25

dsolve(diff(y(x),x$2)+exp(y(x))=0,y(x), singsol=all)
 

\[ y = -\ln \left (2\right )+\ln \left (\frac {\operatorname {sech}\left (\frac {x +c_{2}}{2 c_{1}}\right )^{2}}{c_{1}^{2}}\right ) \]

✓ Solution by Mathematica

Time used: 22.613 (sec). Leaf size: 32

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

\[ y(x)\to \log \left (\frac {1}{2} c_1 \text {sech}^2\left (\frac {1}{2} \sqrt {c_1 (x+c_2){}^2}\right )\right ) \]