Internal problem ID [4846]
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]]
\[ \boxed {{y^{\prime \prime }}^{2}-k^{2} \left ({y^{\prime }}^{2}+1\right )=0} \]
✓ Solution by Maple
Time used: 0.75 (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 \left (x \right ) = -i x +c_{1} y \left (x \right ) = i x +c_{1} y \left (x \right ) = c_{1} +c_{2} {\mathrm e}^{k x}+\frac {{\mathrm e}^{-k x}}{4 k^{2} c_{2}} y \left (x \right ) = 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: 0.451 (sec). Leaf size: 71
DSolve[(y''[x])^2==k^2*(1+ (y'[x])^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {e^{k x-c_1}+e^{-k x+c_1}-2 c_2 k}{2 k} y(x)\to \frac {e^{k x+c_1} \left (1+e^{-2 (k x+c_1)}\right )}{2 k}+c_2 \end{align*}