Internal problem ID [14653]
Internal file name [OUTPUT/14333_Wednesday_April_03_2024_02_17_21_PM_34273/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: 35.
ODE order: 5.
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]]
\[ \boxed {y^{\left (5\right )}+3 y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }+y^{\prime \prime }=0} \] The characteristic equation is \[ \lambda ^{5}+3 \lambda ^{4}+3 \lambda ^{3}+\lambda ^{2} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= -1\\ \lambda _4 &= -1\\ \lambda _5 &= -1 \end {align*}
Therefore the homogeneous solution is \[ y_h(t)=c_{1} {\mathrm e}^{-t}+c_{2} t \,{\mathrm e}^{-t}+t^{2} {\mathrm e}^{-t} c_{3} +c_{4} +t c_{5} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-t}\\ y_2 &= t \,{\mathrm e}^{-t}\\ y_3 &= t^{2} {\mathrm e}^{-t}\\ y_4 &= 1\\ y_5 &= t \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{-t}+c_{2} t \,{\mathrm e}^{-t}+t^{2} {\mathrm e}^{-t} c_{3} +c_{4} +t c_{5} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{-t}+c_{2} t \,{\mathrm e}^{-t}+t^{2} {\mathrm e}^{-t} c_{3} +c_{4} +t c_{5} \] 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: 0.0 (sec). Leaf size: 24
dsolve(diff(y(t),t$5)+3*diff(y(t),t$4)+3*diff(y(t),t$3)+diff(y(t),t$2)=0,y(t), singsol=all)
\[ y \left (t \right ) = \left (c_{5} t^{2}+t c_{4} +c_{3} \right ) {\mathrm e}^{-t}+t c_{2} +c_{1} \]
✓ Solution by Mathematica
Time used: 0.056 (sec). Leaf size: 38
DSolve[y'''''[t]+3*y''''[t]+3*y'''[t]+y''[t]==0,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to e^{-t} \left (c_3 \left (t^2+4 t+6\right )+c_2 (t+2)+c_1\right )+c_5 t+c_4 \]