Internal problem ID [14657]
Internal file name [OUTPUT/14337_Wednesday_April_03_2024_02_17_22_PM_96052526/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.5, page 175
Problem number: 39.
ODE order: 6.
ODE degree: 1.
The type(s) of ODE detected by this program : "higher_order_linear_constant_coefficients_ODE"
Maple gives the following as the ode type
[[_high_order, _missing_x]]
\[ \boxed {y^{\left (6\right )}+12 y^{\prime \prime \prime \prime }+48 y^{\prime \prime }+64 y=0} \] The characteristic equation is \[ \lambda ^{6}+12 \lambda ^{4}+48 \lambda ^{2}+64 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 2 i\\ \lambda _2 &= -2 i\\ \lambda _3 &= 2 i\\ \lambda _4 &= -2 i\\ \lambda _5 &= 2 i\\ \lambda _6 &= -2 i \end {align*}
Therefore the homogeneous solution is \[ y_h(t)={\mathrm e}^{-2 i t} c_{1} +t \,{\mathrm e}^{-2 i t} c_{2} +t^{2} {\mathrm e}^{-2 i t} c_{3} +{\mathrm e}^{2 i t} c_{4} +t \,{\mathrm e}^{2 i t} c_{5} +t^{2} {\mathrm e}^{2 i t} c_{6} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-2 i t}\\ y_2 &= {\mathrm e}^{-2 i t} t\\ y_3 &= t^{2} {\mathrm e}^{-2 i t}\\ y_4 &= {\mathrm e}^{2 i t}\\ y_5 &= {\mathrm e}^{2 i t} t\\ y_6 &= t^{2} {\mathrm e}^{2 i t} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-2 i t} c_{1} +t \,{\mathrm e}^{-2 i t} c_{2} +t^{2} {\mathrm e}^{-2 i t} c_{3} +{\mathrm e}^{2 i t} c_{4} +t \,{\mathrm e}^{2 i t} c_{5} +t^{2} {\mathrm e}^{2 i t} c_{6} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-2 i t} c_{1} +t \,{\mathrm e}^{-2 i t} c_{2} +t^{2} {\mathrm e}^{-2 i t} c_{3} +{\mathrm e}^{2 i t} c_{4} +t \,{\mathrm e}^{2 i t} c_{5} +t^{2} {\mathrm e}^{2 i t} c_{6} \] Verified OK.
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 35
dsolve(diff(y(t),t$6)+12*diff(y(t),t$4)+48*diff(y(t),t$2)+64*y(t)=0,y(t), singsol=all)
\[ y \left (t \right ) = \left (c_{6} t^{2}+t c_{4} +c_{2} \right ) \cos \left (2 t \right )+\sin \left (2 t \right ) \left (c_{5} t^{2}+c_{3} t +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 40
DSolve[y''''''[t]+12*y''''[t]+48*y''[t]+64*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to (t (c_3 t+c_2)+c_1) \cos (2 t)+(t (c_6 t+c_5)+c_4) \sin (2 t) \]