Internal problem ID [11312]
Internal file name [OUTPUT/10298_Tuesday_December_27_2022_04_05_55_AM_57654390/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IX, Miscellaneous methods for solving equations of higher order than first.
Article 57. Dependent variable absent. Page 132
Problem number: Ex 2.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]]
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), y(x)` *** Sublevel 4 *** Methods for high order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature 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))^2 = -x^2+1, y(x), singsol = none` *** Sublevel 2 *** Methods for second order ODEs: Successful isolation of d^2y/dx^2: 2 solutions were found. Trying to solve each resulting ODE. *** Sublevel 3 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful ------------------- * Tackling next ODE. *** Sublevel 3 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 94
dsolve((x*diff(y(x),x$3)-diff(y(x),x$2))^2=diff(y(x),x$3)^2+1,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (x^{2}+2\right ) \sqrt {-x^{2}+1}}{6}+c_{1} x +\frac {x \arcsin \left (x \right )}{2}+c_{2} \\ y \left (x \right ) &= -\frac {x^{2} \sqrt {-x^{2}+1}}{6}-\frac {\sqrt {-x^{2}+1}}{3}-\frac {x \arcsin \left (x \right )}{2}+c_{1} x +c_{2} \\ y \left (x \right ) &= \frac {\sqrt {c_{1}^{2}-1}\, x^{3}}{6}+\frac {c_{1} x^{2}}{2}+c_{2} x +c_{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.241 (sec). Leaf size: 75
DSolve[(x*y'''[x]-y''[x])^2==(y'''[x])^2+1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 x^3}{6}-\frac {1}{2} \sqrt {1+c_1{}^2} x^2+c_3 x+c_2 \\ y(x)\to \frac {c_1 x^3}{6}+\frac {1}{2} \sqrt {1+c_1{}^2} x^2+c_3 x+c_2 \\ \end{align*}