66.2.12 problem Problem 12

Internal problem ID [13840]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 12
Date solved : Monday, March 31, 2025 at 08:14:50 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 }}^{2}+{y^{\prime \prime }}^{2}&=1 \end{align*}

✓ Maple. Time used: 0.223 (sec). Leaf size: 51

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

\begin{align*} y &= -\frac {1}{2} x^{2}+c_{1} x +c_{2} \\ y &= \frac {1}{2} x^{2}+c_{1} x +c_{2} \\ y &= c_{1} +c_{2} x +\sqrt {-c_{3}^{2}+1}\, \sin \left (x \right )+c_{3} \cos \left (x \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.183 (sec). Leaf size: 52

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

\begin{align*} y(x)\to c_3 x+\sin (x-c_1)+c_2 \\ y(x)\to c_3 x-\sin (x+c_1)+c_2 \\ y(x)\to \text {Interval}[\{-1,1\}]+c_3 x+c_2 \\ \end{align*}

✗ Sympy

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

Timed Out