83.35.1 problem 1

Internal problem ID [19342]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Misc. Exercise on chapter VII. Page 118
Problem number : 1
Date solved : Thursday, March 13, 2025 at 02:16:31 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.250 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 60.032 (sec). Leaf size: 32

ode=D[y[x],{x,2}]==Exp[y[x]]; 
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: 15.399 (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)
 

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