53.4.18 problem 19

Internal problem ID [8506]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 19
Date solved : Monday, January 27, 2025 at 04:08:01 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 (3\right )&=0\\ y^{\prime }\left (3\right )&=-1 \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 12

dsolve([diff(y(x),x$2)=-exp(-2*y(x)),y(3) = 0, D(y)(3) = -1],y(x), singsol=all)
 

\[ y = \frac {\ln \left (\left (x -4\right )^{2}\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.162 (sec). Leaf size: 11

DSolve[{D[y[x],{x,2}]==-Exp[-2*y[x]],{y[3]==0,Derivative[1][y][3 ]==-1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \log (4-x) \]