Internal problem ID [12174]
Internal file name [OUTPUT/10827_Thursday_September_21_2023_05_47_33_AM_47536843/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
Unable to solve or complete the solution.
Unable to parse ODE.
Maple trace
`Methods for third order ODEs: *** Sublevel 2 *** Methods for third order ODEs: Successful isolation of d^3y/dx^3: 2 solutions were found. Trying to solve each resulting ODE. *** Sublevel 3 *** Methods for third order ODEs: --- Trying classification methods --- trying 3rd order ODE linearizable_by_differentiation -> Calling odsolve with the ODE`, diff(diff(diff(diff(y(x), x), x), x), x)+diff(diff(y(x), x), x), y(x)` *** Sublevel Methods for high order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful <- 3rd order ODE linearizable_by_differentiation successful ------------------- * Tackling next ODE. *** Sublevel 3 *** Methods for third order ODEs: --- Trying classification methods --- trying 3rd order ODE linearizable_by_differentiation <- 3rd order ODE linearizable_by_differentiation successful -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -1, y(x), singsol = none` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = 1, y(x), singsol = none` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.469 (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 ) &= c_{2} +c_{1} x +\frac {1}{2} x^{2} \\ y \left (x \right ) &= 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: 0.348 (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*}