Internal problem ID [6094]
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: 30.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\left (1+\left (y^{\prime }\right )^{2}\right )^{\frac {3}{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.192 (sec). Leaf size: 51
dsolve(diff(y(x),x$2)=(1+diff(y(x),x)^2)^(3/2),y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x +c_{1} \\ y \relax (x ) = i x +c_{1} \\ y \relax (x ) = \left (c_{1}+x +1\right ) \left (c_{1}+x -1\right ) \sqrt {-\frac {1}{c_{1}^{2}+2 c_{1} x +x^{2}-1}}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.231 (sec). Leaf size: 53
DSolve[y''[x]==(1+(y'[x])^2)^(3/2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-i \sqrt {(x-1+c_1) (x+1+c_1)} \\ y(x)\to i \sqrt {(x-1+c_1) (x+1+c_1)}+c_2 \\ \end{align*}