83.29.6 problem 6

Internal problem ID [19379]
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 : Tuesday, January 28, 2025 at 01:35:55 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*}

✓ Solution by Maple

Time used: 0.549 (sec). Leaf size: 33

dsolve([diff(y(x),x$2)=exp(2*y(x)),y(0) = 0, D(y)(0) = 0],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.084 (sec). Leaf size: 14

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

\[ y(x)\to \frac {1}{2} \log \left (\sec ^2(x)\right ) \]