32.10.19 problem Exercise 35.19, page 504

Internal problem ID [6013]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 8. Special second order equations. Lesson 35. Independent variable x absent
Problem number : Exercise 35.19, page 504
Date solved : Monday, January 27, 2025 at 01:32:13 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 15

dsolve([2*diff(y(x),x$2)=exp(y(x)),y(0) = 0, D(y)(0) = 1],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 15

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

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