35.7.8 problem 5

Internal problem ID [6190]
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
Date solved : Monday, January 27, 2025 at 01:47:10 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} {y^{\prime \prime }}^{2}&=k^{2} \left (1+{y^{\prime }}^{2}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.506 (sec). Leaf size: 53

dsolve((diff(y(x),x$2))^2=k^2*(1+ (diff(y(x),x))^2),y(x), singsol=all)
 

\begin{align*} y &= -i x +c_{1} \\ y &= i x +c_{1} \\ y &= \frac {4 c_{2}^{2} {\mathrm e}^{k x} k^{2}+4 c_{1} c_{2} k^{2}+{\mathrm e}^{-k x}}{4 c_{2} k^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.524 (sec). Leaf size: 40

DSolve[(D[y[x],{x,2}])^2==k^2*(1+ (D[y[x],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*}