Internal problem ID [13489]
Internal file name [OUTPUT/12661_Friday_February_16_2024_12_04_46_AM_80544021/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 13. Higher order equations: Extending first order concepts. Additional exercises
page 259
Problem number: 13.3 (c).
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_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: --- 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)-2, y(x)` *** Sublevel 2 *** Methods for high order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful differential order: 3; trying a linearization to 4th order trying differential order: 3; missing variables `, `-> Computing symmetries using: way = 3 -> Calling odsolve with the ODE`, diff(_b(_a), _a) = 2*_b(_a)^(1/2), _b(_a), HINT = [[1, 0], [_a, 2*_b]]` *** Sublevel 2 *** symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[1, 0], [_a, 2*_b]
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 36
dsolve(diff(y(x),x$3)=2*sqrt(diff(y(x),x$2)),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= c_{1} x +c_{2} \\ y \left (x \right ) &= \frac {1}{12} x^{4}+\frac {1}{3} c_{1} x^{3}+\frac {1}{2} c_{1}^{2} x^{2}+c_{2} x +c_{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.13 (sec). Leaf size: 39
DSolve[y'''[x]==2*Sqrt[y''[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^4}{12}+\frac {c_1 x^3}{6}+\frac {c_1{}^2 x^2}{8}+c_3 x+c_2 \]