56.34.3 problem Ex 3

Internal problem ID [14276]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 58. Independent variable absent. Page 135
Problem number : Ex 3
Date solved : Thursday, October 02, 2025 at 09:27:35 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} 2 y^{\prime \prime }&={\mathrm e}^{y} \end{align*}

✓ Maple. Time used: 0.108 (sec). Leaf size: 20

ode:=2*diff(diff(y(x),x),x) = exp(y(x)); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 49.407 (sec). Leaf size: 30

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

\begin{align*} y(x)&\to \log \left (-c_1 \text {sech}^2\left (\frac {1}{2} \sqrt {c_1 (x+c_2){}^2}\right )\right ) \end{align*}

✓ Sympy. Time used: 41.716 (sec). Leaf size: 44

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

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