77.29.6 problem 6

Internal problem ID [20664]
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. Exercise VII (C) at page 107
Problem number : 6
Date solved : Thursday, October 02, 2025 at 06:18:10 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.473 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x) = exp(2*y(x)); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\begin{align*} y &= \operatorname {RootOf}\left (-\arctan \left (\sqrt {{\mathrm e}^{2 \textit {\_Z}}-1}\right )+x \right ) \\ y &= \operatorname {RootOf}\left (\arctan \left (\sqrt {{\mathrm e}^{2 \textit {\_Z}}-1}\right )+x \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 14

ode=D[y[x],{x,2}]==Exp[2*y[x]]; 
ic={y[0]==0,Derivative[1][y][0] == 0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \log \left (\sec ^2(x)\right ) \end{align*}

✗ Sympy

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

NotImplementedError : Initial conditions produced too many solutions for constants