54.7.14 problem 1604 (6.14)

Internal problem ID [12863]
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 : Wednesday, October 01, 2025 at 02:22:45 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.171 (sec). Leaf size: 25

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

\[ y = -\ln \left (2\right )+\ln \left (\frac {\sec \left (\frac {c_2 +x}{2 c_1}\right )^{2}}{c_1^{2}}\right ) \]

✓ Mathematica. Time used: 48.004 (sec). Leaf size: 32

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

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 9.499 (sec). Leaf size: 41

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

\[ \left [ y{\left (x \right )} = \log {\left (- \frac {C_{1}}{\cos {\left (\sqrt {- C_{1}} \left (C_{2} + x\right ) \right )} + 1} \right )}, \ y{\left (x \right )} = \log {\left (- \frac {C_{1}}{\cos {\left (\sqrt {- C_{1}} \left (C_{2} - x\right ) \right )} + 1} \right )}\right ] \]