Internal problem ID [4338]
Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley.
2006
Section: Chapter 8, Ordinary differential equations. Section 7. Other second-Order equations. page
435
Problem number: 5.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {\left (y^{\prime \prime }\right )^{2}-k^{2} \left (\left (y^{\prime }\right )^{2}+1\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 1.61 (sec). Leaf size: 67
dsolve((diff(y(x),x$2))^2=k^2*(1+ (diff(y(x),x))^2),y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x +c_{1} \\ y \relax (x ) = i x +c_{1} \\ y \relax (x ) = c_{1}+c_{2} {\mathrm e}^{k x}+\frac {{\mathrm e}^{-k x}}{4 k^{2} c_{2}} \\ y \relax (x ) = c_{1}+\frac {{\mathrm e}^{k x}}{4 c_{2} k^{2}}+c_{2} {\mathrm e}^{-k x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.728 (sec). Leaf size: 40
DSolve[(y''[x])^2==k^2*(1+ (y'[x])^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-\frac {\cosh (k x-c_1)}{k} \\ y(x)\to \frac {\cosh (k x+c_1)}{k}+c_2 \\ \end{align*}