Internal problem ID [15207]
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 14. Differential equations admitting of
depression of their order. Exercises page 107
Problem number: 349.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {3 y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}}=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 49
dsolve(3*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 +3\right ) \left (c_{1} +x -3\right ) \sqrt {-\frac {1}{\left (c_{1} +x +3\right ) \left (c_{1} +x -3\right )}}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.26 (sec). Leaf size: 63
DSolve[3*y''[x]==(1+y'[x]^2)^(3/2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-i \sqrt {x^2+6 c_1 x-9+9 c_1{}^2} \\ y(x)\to i \sqrt {x^2+6 c_1 x-9+9 c_1{}^2}+c_2 \\ \end{align*}