Internal problem ID [6085]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+{\mathrm e}^{-2 y}=0} \end {gather*} With initial conditions \begin {align*} [y \relax (3) = 0, y^{\prime }\relax (3) = -1] \end {align*}
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \relax (x ) = \frac {\ln \left (\left (x -4\right )^{2}\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.241 (sec). Leaf size: 11
DSolve[{y''[x]==-Exp[-2*y[x]],{y[3]==0,y'[3]==-1}},y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \log (4-x) \\ \end{align*}