Internal problem ID [14698]
Internal file name [OUTPUT/14378_Wednesday_April_03_2024_02_17_50_PM_85726512/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.6, page 187
Problem number: 24.
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]]
\[ \boxed {y^{\left (6\right )}+y^{\prime \prime \prime \prime }=-24} \] This is higher order nonhomogeneous ODE. Let the solution be \[ y = y_h + y_p \] Where \(y_h\) is the solution to the homogeneous ODE And \(y_p\) is a particular solution to the nonhomogeneous ODE. \(y_h\) is the solution to \[ y^{\left (6\right )}+y^{\prime \prime \prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{6}+\lambda ^{4} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= 0\\ \lambda _4 &= 0\\ \lambda _5 &= i\\ \lambda _6 &= -i \end {align*}
Therefore the homogeneous solution is \[ y_h(t)=t^{3} c_{4} +t^{2} c_{3} +c_{2} t +c_{1} +{\mathrm e}^{i t} c_{5} +{\mathrm e}^{-i t} c_{6} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= t \\ y_3 &= t^{2} \\ y_4 &= t^{3} \\ y_5 &= {\mathrm e}^{i t} \\ y_6 &= {\mathrm e}^{-i t} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\left (6\right )}+y^{\prime \prime \prime \prime } = -24 \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 1 \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, t, t^{2}, t^{3}, {\mathrm e}^{i t}, {\mathrm e}^{-i t}\} \] Since \(1\) is duplicated in the UC_set, then this basis is multiplied by extra \(t\). The UC_set becomes \[ [\{t\}] \] Since \(t\) is duplicated in the UC_set, then this basis is multiplied by extra \(t\). The UC_set becomes \[ [\{t^{2}\}] \] Since \(t^{2}\) is duplicated in the UC_set, then this basis is multiplied by extra \(t\). The UC_set becomes \[ [\{t^{3}\}] \] Since \(t^{3}\) is duplicated in the UC_set, then this basis is multiplied by extra \(t\). The UC_set becomes \[ [\{t^{4}\}] \] Since there was duplication between the basis functions in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis function in the above updated UC_set. \[ y_p = A_{1} t^{4} \] The unknowns \(\{A_{1}\}\) are found by substituting the above trial solution \(y_p\) into the ODE and comparing coefficients. Substituting the trial solution into the ODE and simplifying gives \[ 24 A_{1} = -24 \] Solving for the unknowns by comparing coefficients results in \[ [A_{1} = -1] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = -t^{4} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (t^{3} c_{4} +t^{2} c_{3} +c_{2} t +c_{1} +{\mathrm e}^{i t} c_{5} +{\mathrm e}^{-i t} c_{6}\right ) + \left (-t^{4}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= t^{3} c_{4} +t^{2} c_{3} +c_{2} t +c_{1} +{\mathrm e}^{i t} c_{5} +{\mathrm e}^{-i t} c_{6} -t^{4} \\ \end{align*}
Verification of solutions
\[ y = t^{3} c_{4} +t^{2} c_{3} +c_{2} t +c_{1} +{\mathrm e}^{i t} c_{5} +{\mathrm e}^{-i t} c_{6} -t^{4} \] Verified OK.
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 6; linear nonhomogeneous with symmetry [0,1] -> Calling odsolve with the ODE`, diff(diff(_b(_a), _a), _a) = -_b(_a)-24, _b(_a)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] -> Try solving first the homogeneous part of the ODE checking if the LODE has constant coefficients <- constant coefficients successful <- solving first the homogeneous part of the ODE successful <- differential order: 6; linear nonhomogeneous with symmetry [0,1] successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 34
dsolve(diff(y(t),t$6)+diff(y(t),t$4)=-24,y(t), singsol=all)
\[ y \left (t \right ) = \frac {c_{3} t^{3}}{6}-t^{4}+\frac {c_{4} t^{2}}{2}+\cos \left (t \right ) c_{1} +\sin \left (t \right ) c_{2} +c_{5} t +c_{6} \]
✓ Solution by Mathematica
Time used: 0.149 (sec). Leaf size: 39
DSolve[y''''''[t]+y''''[t]==-24,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to -t^4+c_6 t^3+c_5 t^2+c_4 t+c_1 \cos (t)+c_2 \sin (t)+c_3 \]