75.14.17 problem 343

Internal problem ID [16931]
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 ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 343
Date solved : Tuesday, January 28, 2025 at 09:41:13 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }&=\sqrt {1-{y^{\prime }}^{2}} \end{align*}

✓ Solution by Maple

Time used: 3.488 (sec). Leaf size: 26

dsolve(diff(y(x),x$2)=sqrt(1-diff(y(x),x)^2),y(x), singsol=all)
 

\begin{align*} y &= -x +c_{1} \\ y &= x +c_{1} \\ y &= -\cos \left (x +c_{1} \right )+c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 54.755 (sec). Leaf size: 60

DSolve[D[y[x],{x,2}]==Sqrt[1-D[y[x],x]^2],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \int _1^x\sin (c_1+K[1])dK[1]+c_2 \\ y(x)\to -\cos \left (2 \pi \text {frac}\left (\frac {x-1}{2 \pi }\right )+1\right )+\cos (1)+c_2 \\ y(x)\to \text {Interval}[\{-2,2\}]+c_2 \\ \end{align*}