Internal problem ID [15183]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 13. Basic concepts and definitions. Exercises
page 98
Problem number: 325.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}}=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 49
dsolve(diff(y(x),x$2)=(1+diff(y(x),x)^2)^(3/2),y(x), singsol=all)
\begin{align*} y &= -i x +c_{1} \\ y &= i x +c_{1} \\ y &= \left (c_{1} +x +1\right ) \left (c_{1} +x -1\right ) \sqrt {-\frac {1}{\left (c_{1} +x +1\right ) \left (c_{1} +x -1\right )}}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.253 (sec). Leaf size: 59
DSolve[y''[x]==(1+y'[x]^2)^(3/2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-i \sqrt {x^2+2 c_1 x-1+c_1{}^2} \\ y(x)\to i \sqrt {x^2+2 c_1 x-1+c_1{}^2}+c_2 \\ \end{align*}