66.2.12 problem Problem 12

Internal problem ID [13911]
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 : Tuesday, January 28, 2025 at 06:08:17 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*}

✓ Solution by Maple

Time used: 0.865 (sec). Leaf size: 51

dsolve(diff(y(x),x$3)^2+diff(y(x),x$2)^2=1,y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[D[y[x],{x,3}]^2+D[y[x],{x,2}]^2==1,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*}