60.7.14 problem 1604 (6.14)

Internal problem ID [11603]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1604 (6.14)
Date solved : Monday, January 27, 2025 at 11:24:06 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: 0.776 (sec). Leaf size: 25

dsolve(diff(diff(y(x),x),x)-exp(y(x))=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 60.038 (sec). Leaf size: 32

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

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