75.14.13 problem 339

Internal problem ID [16848]
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 : 339
Date solved : Thursday, March 13, 2025 at 08:56:51 AM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]

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

✗ Maple

ode:=diff(diff(diff(y(x),x),x),x) = (1-diff(diff(y(x),x),x)^2)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 60.072 (sec). Leaf size: 33

ode=D[y[x],{x,3}]==Sqrt[1-D[y[x],{x,2}]^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \int _1^x\int _1^{K[2]}\sin (c_1+K[1])dK[1]dK[2]+c_3 x+c_2 \]

✓ Sympy. Time used: 0.300 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt(1 - Derivative(y(x), (x, 2))**2) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + C_{2} x - \sin {\left (C_{3} + x \right )} \]