Internal problem ID [10827]
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.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
✓ Solution by Maple
Time used: 0.187 (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 \left (x \right ) = -\frac {1}{2} x^{2}+c_{1} x +c_{2} \\ y \left (x \right ) = \frac {1}{2} x^{2}+c_{1} x +c_{2} \\ y \left (x \right ) = c_{1} +x c_{2} +\sqrt {-c_{3}^{2}+1}\, \sin \left (x \right )+c_{3} \cos \left (x \right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.244 (sec). Leaf size: 54
DSolve[y'''[x]^2+y''[x]^2==1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_3 x-\cos (x-c_1)+c_2 \\ y(x)\to c_3 x-\cos (x+c_1)+c_2 \\ y(x)\to \text {Interval}[\{-1,1\}]+c_3 x+c_2 \\ \end{align*}