56.5.22 problem 22

Internal problem ID [8983]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 22
Date solved : Sunday, March 30, 2025 at 01:57:56 PM
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: 0.026 (sec). Leaf size: 25

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

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

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

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

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

✓ Sympy. Time used: 15.788 (sec). Leaf size: 37

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)
 

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