14.8 problem 8

Internal problem ID [14682]
Internal file name [OUTPUT/14362_Wednesday_April_03_2024_02_17_40_PM_868064/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: 8.
ODE order: 4.
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, _with_linear_symmetries]]

\[ \boxed {y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+13 y^{\prime \prime }-24 y^{\prime }+36 y=108 t} \] 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 \[ y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+13 y^{\prime \prime }-24 y^{\prime }+36 y = 0 \] The characteristic equation is \[ \lambda ^{4}-6 \lambda ^{3}+13 \lambda ^{2}-24 \lambda +36 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 2 i\\ \lambda _2 &= -2 i\\ \lambda _3 &= 3\\ \lambda _4 &= 3 \end {align*}

Therefore the homogeneous solution is \[ y_h(t)=c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{3 t} t +{\mathrm e}^{-2 i t} c_{3} +{\mathrm e}^{2 i t} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{3 t} \\ y_2 &= {\mathrm e}^{3 t} t \\ y_3 &= {\mathrm e}^{-2 i t} \\ y_4 &= {\mathrm e}^{2 i t} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+13 y^{\prime \prime }-24 y^{\prime }+36 y = 108 t \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ t \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1, t\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{{\mathrm e}^{3 t} t, {\mathrm e}^{3 t}, {\mathrm e}^{-2 i t}, {\mathrm e}^{2 i t}\} \] Since there is no duplication between the basis function in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis in the UC_set. \[ y_p = A_{2} t +A_{1} \] The unknowns \(\{A_{1}, A_{2}\}\) 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 \[ 36 A_{2} t +36 A_{1}-24 A_{2} = 108 t \] Solving for the unknowns by comparing coefficients results in \[ [A_{1} = 2, A_{2} = 3] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = 3 t +2 \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{3 t} t +{\mathrm e}^{-2 i t} c_{3} +{\mathrm e}^{2 i t} c_{4}\right ) + \left (3 t +2\right ) \\ \end{align*} Which simplifies to \[ y = {\mathrm e}^{-2 i t} c_{3} +{\mathrm e}^{2 i t} c_{4} +{\mathrm e}^{3 t} \left (c_{2} t +c_{1} \right )+3 t +2 \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-2 i t} c_{3} +{\mathrm e}^{2 i t} c_{4} +{\mathrm e}^{3 t} \left (c_{2} t +c_{1} \right )+3 t +2 \\ \end{align*}

Verification of solutions

\[ y = {\mathrm e}^{-2 i t} c_{3} +{\mathrm e}^{2 i t} c_{4} +{\mathrm e}^{3 t} \left (c_{2} t +c_{1} \right )+3 t +2 \] Verified OK.

14.8.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+13 y^{\prime \prime }-24 y^{\prime }+36 y=108 t \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & y^{\prime \prime \prime \prime } \\ \square & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{1}\left (t \right ) \\ {} & {} & y_{1}\left (t \right )=y \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{2}\left (t \right ) \\ {} & {} & y_{2}\left (t \right )=y^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{3}\left (t \right ) \\ {} & {} & y_{3}\left (t \right )=y^{\prime \prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{4}\left (t \right ) \\ {} & {} & y_{4}\left (t \right )=y^{\prime \prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{4}^{\prime }\left (t \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{4}^{\prime }\left (t \right )=108 t +6 y_{4}\left (t \right )-13 y_{3}\left (t \right )+24 y_{2}\left (t \right )-36 y_{1}\left (t \right ) \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [y_{2}\left (t \right )=y_{1}^{\prime }\left (t \right ), y_{3}\left (t \right )=y_{2}^{\prime }\left (t \right ), y_{4}\left (t \right )=y_{3}^{\prime }\left (t \right ), y_{4}^{\prime }\left (t \right )=108 t +6 y_{4}\left (t \right )-13 y_{3}\left (t \right )+24 y_{2}\left (t \right )-36 y_{1}\left (t \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (t \right )=\left [\begin {array}{c} y_{1}\left (t \right ) \\ y_{2}\left (t \right ) \\ y_{3}\left (t \right ) \\ y_{4}\left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (t \right )=\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -36 & 24 & -13 & 6 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (t \right )+\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 108 t \end {array}\right ] \\ \bullet & {} & \textrm {Define the forcing function}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{f}}\left (t \right )=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 108 t \end {array}\right ] \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -36 & 24 & -13 & 6 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (t \right )=A \cdot {\moverset {\rightarrow }{y}}\left (t \right )+{\moverset {\rightarrow }{f}} \\ \bullet & {} & \textrm {To solve the system, find the eigenvalues and eigenvectors of}\hspace {3pt} A \\ \bullet & {} & \textrm {Eigenpairs of}\hspace {3pt} A \\ {} & {} & \left [\left [3, \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]\right ], \left [3, \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ], \left [-2 \,\mathrm {I}, \left [\begin {array}{c} -\frac {\mathrm {I}}{8} \\ -\frac {1}{4} \\ \frac {\mathrm {I}}{2} \\ 1 \end {array}\right ]\right ], \left [2 \,\mathrm {I}, \left [\begin {array}{c} \frac {\mathrm {I}}{8} \\ -\frac {1}{4} \\ -\frac {\mathrm {I}}{2} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair, with eigenvalue of algebraic multiplicity 2}\hspace {3pt} \\ {} & {} & \left [3, \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {First solution from eigenvalue}\hspace {3pt} 3 \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}\left (t \right )={\mathrm e}^{3 t}\cdot \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Form of the 2nd homogeneous solution where}\hspace {3pt} {\moverset {\rightarrow }{p}}\hspace {3pt}\textrm {is to be solved for,}\hspace {3pt} \lambda =3\hspace {3pt}\textrm {is the eigenvalue, and}\hspace {3pt} {\moverset {\rightarrow }{v}}\hspace {3pt}\textrm {is the eigenvector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}\left (t \right )={\mathrm e}^{\lambda t} \left (t {\moverset {\rightarrow }{v}}+{\moverset {\rightarrow }{p}}\right ) \\ \bullet & {} & \textrm {Note that the}\hspace {3pt} t \hspace {3pt}\textrm {multiplying}\hspace {3pt} {\moverset {\rightarrow }{v}}\hspace {3pt}\textrm {makes this solution linearly independent to the 1st solution obtained from}\hspace {3pt} \lambda =3 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} {\moverset {\rightarrow }{y}}_{2}\left (t \right )\hspace {3pt}\textrm {into the homogeneous system}\hspace {3pt} \\ {} & {} & \lambda \,{\mathrm e}^{\lambda t} \left (t {\moverset {\rightarrow }{v}}+{\moverset {\rightarrow }{p}}\right )+{\mathrm e}^{\lambda t} {\moverset {\rightarrow }{v}}=\left ({\mathrm e}^{\lambda t} A \right )\cdot \left (t {\moverset {\rightarrow }{v}}+{\moverset {\rightarrow }{p}}\right ) \\ \bullet & {} & \textrm {Use the fact that}\hspace {3pt} {\moverset {\rightarrow }{v}}\hspace {3pt}\textrm {is an eigenvector of}\hspace {3pt} A \\ {} & {} & \lambda \,{\mathrm e}^{\lambda t} \left (t {\moverset {\rightarrow }{v}}+{\moverset {\rightarrow }{p}}\right )+{\mathrm e}^{\lambda t} {\moverset {\rightarrow }{v}}={\mathrm e}^{\lambda t} \left (\lambda t {\moverset {\rightarrow }{v}}+A \cdot {\moverset {\rightarrow }{p}}\right ) \\ \bullet & {} & \textrm {Simplify equation}\hspace {3pt} \\ {} & {} & \lambda {\moverset {\rightarrow }{p}}+{\moverset {\rightarrow }{v}}=A \cdot {\moverset {\rightarrow }{p}} \\ \bullet & {} & \textrm {Make use of the identity matrix}\hspace {3pt} \mathrm {I} \\ {} & {} & \left (\lambda \cdot I \right )\cdot {\moverset {\rightarrow }{p}}+{\moverset {\rightarrow }{v}}=A \cdot {\moverset {\rightarrow }{p}} \\ \bullet & {} & \textrm {Condition}\hspace {3pt} {\moverset {\rightarrow }{p}}\hspace {3pt}\textrm {must meet for}\hspace {3pt} {\moverset {\rightarrow }{y}}_{2}\left (t \right )\hspace {3pt}\textrm {to be a solution to the homogeneous system}\hspace {3pt} \\ {} & {} & \left (A -\lambda \cdot I \right )\cdot {\moverset {\rightarrow }{p}}={\moverset {\rightarrow }{v}} \\ \bullet & {} & \textrm {Choose}\hspace {3pt} {\moverset {\rightarrow }{p}}\hspace {3pt}\textrm {to use in the second solution to the homogeneous system from eigenvalue}\hspace {3pt} 3 \\ {} & {} & \left (\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -36 & 24 & -13 & 6 \end {array}\right ]-3\cdot \left [\begin {array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end {array}\right ]\right )\cdot {\moverset {\rightarrow }{p}}=\left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Choice of}\hspace {3pt} {\moverset {\rightarrow }{p}} \\ {} & {} & {\moverset {\rightarrow }{p}}=\left [\begin {array}{c} -\frac {1}{81} \\ 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {Second solution from eigenvalue}\hspace {3pt} 3 \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}\left (t \right )={\mathrm e}^{3 t}\cdot \left (t \cdot \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]+\left [\begin {array}{c} -\frac {1}{81} \\ 0 \\ 0 \\ 0 \end {array}\right ]\right ) \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-2 \,\mathrm {I}, \left [\begin {array}{c} -\frac {\mathrm {I}}{8} \\ -\frac {1}{4} \\ \frac {\mathrm {I}}{2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-2 \,\mathrm {I} t}\cdot \left [\begin {array}{c} -\frac {\mathrm {I}}{8} \\ -\frac {1}{4} \\ \frac {\mathrm {I}}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Use Euler identity to write solution in terms of}\hspace {3pt} \sin \hspace {3pt}\textrm {and}\hspace {3pt} \cos \\ {} & {} & \left (\cos \left (2 t \right )-\mathrm {I} \sin \left (2 t \right )\right )\cdot \left [\begin {array}{c} -\frac {\mathrm {I}}{8} \\ -\frac {1}{4} \\ \frac {\mathrm {I}}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & \left [\begin {array}{c} -\frac {\mathrm {I}}{8} \left (\cos \left (2 t \right )-\mathrm {I} \sin \left (2 t \right )\right ) \\ -\frac {\cos \left (2 t \right )}{4}+\frac {\mathrm {I} \sin \left (2 t \right )}{4} \\ \frac {\mathrm {I}}{2} \left (\cos \left (2 t \right )-\mathrm {I} \sin \left (2 t \right )\right ) \\ \cos \left (2 t \right )-\mathrm {I} \sin \left (2 t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{3}\left (t \right )=\left [\begin {array}{c} -\frac {\sin \left (2 t \right )}{8} \\ -\frac {\cos \left (2 t \right )}{4} \\ \frac {\sin \left (2 t \right )}{2} \\ \cos \left (2 t \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (t \right )=\left [\begin {array}{c} -\frac {\cos \left (2 t \right )}{8} \\ \frac {\sin \left (2 t \right )}{4} \\ \frac {\cos \left (2 t \right )}{2} \\ -\sin \left (2 t \right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {General solution of the system of ODEs can be written in terms of the particular solution}\hspace {3pt} {\moverset {\rightarrow }{y}}_{p}\left (t \right ) \\ {} & {} & {\moverset {\rightarrow }{y}}\left (t \right )=c_{1} {\moverset {\rightarrow }{y}}_{1}\left (t \right )+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (t \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (t \right )+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (t \right )+{\moverset {\rightarrow }{y}}_{p}\left (t \right ) \\ \square & {} & \textrm {Fundamental matrix}\hspace {3pt} \\ {} & \circ & \textrm {Let}\hspace {3pt} \phi \left (t \right )\hspace {3pt}\textrm {be the matrix whose columns are the independent solutions of the homogeneous system.}\hspace {3pt} \\ {} & {} & \phi \left (t \right )=\left [\begin {array}{cccc} \frac {{\mathrm e}^{3 t}}{27} & {\mathrm e}^{3 t} \left (\frac {t}{27}-\frac {1}{81}\right ) & -\frac {\sin \left (2 t \right )}{8} & -\frac {\cos \left (2 t \right )}{8} \\ \frac {{\mathrm e}^{3 t}}{9} & \frac {{\mathrm e}^{3 t} t}{9} & -\frac {\cos \left (2 t \right )}{4} & \frac {\sin \left (2 t \right )}{4} \\ \frac {{\mathrm e}^{3 t}}{3} & \frac {{\mathrm e}^{3 t} t}{3} & \frac {\sin \left (2 t \right )}{2} & \frac {\cos \left (2 t \right )}{2} \\ {\mathrm e}^{3 t} & {\mathrm e}^{3 t} t & \cos \left (2 t \right ) & -\sin \left (2 t \right ) \end {array}\right ] \\ {} & \circ & \textrm {The fundamental matrix,}\hspace {3pt} \Phi \left (t \right )\hspace {3pt}\textrm {is a normalized version of}\hspace {3pt} \phi \left (t \right )\hspace {3pt}\textrm {satisfying}\hspace {3pt} \Phi \left (0\right )=I \hspace {3pt}\textrm {where}\hspace {3pt} I \hspace {3pt}\textrm {is the identity matrix}\hspace {3pt} \\ {} & {} & \Phi \left (t \right )=\phi \left (t \right )\cdot \frac {1}{\phi \left (0\right )} \\ {} & \circ & \textrm {Substitute the value of}\hspace {3pt} \phi \left (t \right )\hspace {3pt}\textrm {and}\hspace {3pt} \phi \left (0\right ) \\ {} & {} & \Phi \left (t \right )=\left [\begin {array}{cccc} \frac {{\mathrm e}^{3 t}}{27} & {\mathrm e}^{3 t} \left (\frac {t}{27}-\frac {1}{81}\right ) & -\frac {\sin \left (2 t \right )}{8} & -\frac {\cos \left (2 t \right )}{8} \\ \frac {{\mathrm e}^{3 t}}{9} & \frac {{\mathrm e}^{3 t} t}{9} & -\frac {\cos \left (2 t \right )}{4} & \frac {\sin \left (2 t \right )}{4} \\ \frac {{\mathrm e}^{3 t}}{3} & \frac {{\mathrm e}^{3 t} t}{3} & \frac {\sin \left (2 t \right )}{2} & \frac {\cos \left (2 t \right )}{2} \\ {\mathrm e}^{3 t} & {\mathrm e}^{3 t} t & \cos \left (2 t \right ) & -\sin \left (2 t \right ) \end {array}\right ]\cdot \frac {1}{\left [\begin {array}{cccc} \frac {1}{27} & -\frac {1}{81} & 0 & -\frac {1}{8} \\ \frac {1}{9} & 0 & -\frac {1}{4} & 0 \\ \frac {1}{3} & 0 & 0 & \frac {1}{2} \\ 1 & 0 & 1 & 0 \end {array}\right ]} \\ {} & \circ & \textrm {Evaluate and simplify to get the fundamental matrix}\hspace {3pt} \\ {} & {} & \Phi \left (t \right )=\left [\begin {array}{cccc} \left (-3 t +1\right ) {\mathrm e}^{3 t} & {\mathrm e}^{3 t} t +\frac {3 \cos \left (2 t \right )}{13}+\frac {9 \sin \left (2 t \right )}{26}-\frac {3 \,{\mathrm e}^{3 t}}{13} & \frac {\left (-3 t +1\right ) {\mathrm e}^{3 t}}{4}-\frac {\cos \left (2 t \right )}{4} & \frac {\left (13 t -3\right ) {\mathrm e}^{3 t}}{52}+\frac {3 \cos \left (2 t \right )}{52}-\frac {\sin \left (2 t \right )}{26} \\ -9 \,{\mathrm e}^{3 t} t & \frac {4 \,{\mathrm e}^{3 t}}{13}+3 \,{\mathrm e}^{3 t} t +\frac {9 \cos \left (2 t \right )}{13}-\frac {6 \sin \left (2 t \right )}{13} & -\frac {9 \,{\mathrm e}^{3 t} t}{4}+\frac {\sin \left (2 t \right )}{2} & \frac {\left (4+39 t \right ) {\mathrm e}^{3 t}}{52}-\frac {\cos \left (2 t \right )}{13}-\frac {3 \sin \left (2 t \right )}{26} \\ -27 \,{\mathrm e}^{3 t} t & \frac {12 \,{\mathrm e}^{3 t}}{13}+9 \,{\mathrm e}^{3 t} t -\frac {18 \sin \left (2 t \right )}{13}-\frac {12 \cos \left (2 t \right )}{13} & -\frac {27 \,{\mathrm e}^{3 t} t}{4}+\cos \left (2 t \right ) & \frac {3 \left (4+39 t \right ) {\mathrm e}^{3 t}}{52}-\frac {3 \cos \left (2 t \right )}{13}+\frac {2 \sin \left (2 t \right )}{13} \\ -81 \,{\mathrm e}^{3 t} t & \frac {36 \,{\mathrm e}^{3 t}}{13}+27 \,{\mathrm e}^{3 t} t -\frac {36 \cos \left (2 t \right )}{13}+\frac {24 \sin \left (2 t \right )}{13} & -\frac {81 \,{\mathrm e}^{3 t} t}{4}-2 \sin \left (2 t \right ) & \frac {9 \left (4+39 t \right ) {\mathrm e}^{3 t}}{52}+\frac {4 \cos \left (2 t \right )}{13}+\frac {6 \sin \left (2 t \right )}{13} \end {array}\right ] \\ \square & {} & \textrm {Find a particular solution of the system of ODEs using variation of parameters}\hspace {3pt} \\ {} & \circ & \textrm {Let the particular solution be the fundamental matrix multiplied by}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (t \right )\hspace {3pt}\textrm {and solve for}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (t \right ) \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (t \right )=\Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right ) \\ {} & \circ & \textrm {Take the derivative of the particular solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}^{\prime }\left (t \right )=\Phi ^{\prime }\left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+\Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (t \right ) \\ {} & \circ & \textrm {Substitute particular solution and its derivative into the system of ODEs}\hspace {3pt} \\ {} & {} & \Phi ^{\prime }\left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+\Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (t \right )=A \cdot \Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+{\moverset {\rightarrow }{f}}\left (t \right ) \\ {} & \circ & \textrm {The fundamental matrix has columns that are solutions to the homogeneous system so its derivative follows that of the homogeneous system}\hspace {3pt} \\ {} & {} & A \cdot \Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+\Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (t \right )=A \cdot \Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}\left (t \right )+{\moverset {\rightarrow }{f}}\left (t \right ) \\ {} & \circ & \textrm {Cancel like terms}\hspace {3pt} \\ {} & {} & \Phi \left (t \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (t \right )={\moverset {\rightarrow }{f}}\left (t \right ) \\ {} & \circ & \textrm {Multiply by the inverse of the fundamental matrix}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{v}}^{\prime }\left (t \right )=\frac {1}{\Phi \left (t \right )}\cdot {\moverset {\rightarrow }{f}}\left (t \right ) \\ {} & \circ & \textrm {Integrate to solve for}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (t \right ) \\ {} & {} & {\moverset {\rightarrow }{v}}\left (t \right )=\int _{0}^{t}\frac {1}{\Phi \left (s \right )}\cdot {\moverset {\rightarrow }{f}}\left (s \right )d s \\ {} & \circ & \textrm {Plug}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (t \right )\hspace {3pt}\textrm {into the equation for the particular solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (t \right )=\Phi \left (t \right )\cdot \left (\int _{0}^{t}\frac {1}{\Phi \left (s \right )}\cdot {\moverset {\rightarrow }{f}}\left (s \right )d s \right ) \\ {} & \circ & \textrm {Plug in the fundamental matrix and the forcing function and compute}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (t \right )=\left [\begin {array}{c} 3 \,{\mathrm e}^{3 t} t -\frac {81 \cos \left (t \right )^{2}}{26}+\frac {27 \cos \left (t \right ) \sin \left (t \right )}{13}-\frac {35 \,{\mathrm e}^{3 t}}{13}+3 t +\frac {151}{26} \\ 3+\frac {27 \cos \left (2 t \right )}{13}+\frac {81 \sin \left (2 t \right )}{26}-\frac {66 \,{\mathrm e}^{3 t}}{13}+9 \,{\mathrm e}^{3 t} t \\ -\frac {108 \cos \left (t \right ) \sin \left (t \right )}{13}+27 \,{\mathrm e}^{3 t} t +\frac {162 \cos \left (t \right )^{2}}{13}-\frac {198 \,{\mathrm e}^{3 t}}{13}+27 t +\frac {36}{13} \\ -\frac {324 \cos \left (t \right ) \sin \left (t \right )}{13}+81 \,{\mathrm e}^{3 t} t -\frac {216 \cos \left (t \right )^{2}}{13}-\frac {594 \,{\mathrm e}^{3 t}}{13}+81 t +\frac {810}{13} \end {array}\right ] \\ \bullet & {} & \textrm {Plug particular solution back into general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (t \right )=c_{1} {\moverset {\rightarrow }{y}}_{1}\left (t \right )+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (t \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (t \right )+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (t \right )+\left [\begin {array}{c} 3 \,{\mathrm e}^{3 t} t -\frac {81 \cos \left (t \right )^{2}}{26}+\frac {27 \cos \left (t \right ) \sin \left (t \right )}{13}-\frac {35 \,{\mathrm e}^{3 t}}{13}+3 t +\frac {151}{26} \\ 3+\frac {27 \cos \left (2 t \right )}{13}+\frac {81 \sin \left (2 t \right )}{26}-\frac {66 \,{\mathrm e}^{3 t}}{13}+9 \,{\mathrm e}^{3 t} t \\ -\frac {108 \cos \left (t \right ) \sin \left (t \right )}{13}+27 \,{\mathrm e}^{3 t} t +\frac {162 \cos \left (t \right )^{2}}{13}-\frac {198 \,{\mathrm e}^{3 t}}{13}+27 t +\frac {36}{13} \\ -\frac {324 \cos \left (t \right ) \sin \left (t \right )}{13}+81 \,{\mathrm e}^{3 t} t -\frac {216 \cos \left (t \right )^{2}}{13}-\frac {594 \,{\mathrm e}^{3 t}}{13}+81 t +\frac {810}{13} \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {151}{26}+\frac {\left (-2835+39 \left (81+c_{2} \right ) t +39 c_{1} -13 c_{2} \right ) {\mathrm e}^{3 t}}{1053}+\frac {\left (-162-13 c_{4} \right ) \cos \left (t \right )^{2}}{52}+\frac {\left (108-13 c_{3} \right ) \sin \left (t \right ) \cos \left (t \right )}{52}+3 t +\frac {c_{4}}{8} \end {array} \]

Maple trace

`Methods for high order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 4; linear nonhomogeneous with symmetry [0,1] 
trying high order linear exact nonhomogeneous 
trying differential order: 4; missing the dependent variable 
checking if the LODE has constant coefficients 
<- constant coefficients successful`
 

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 31

dsolve(diff(y(t),t$4)-6*diff(y(t),t$3)+13*diff(y(t),t$2)-24*diff(y(t),t)+36*y(t)=108*t,y(t), singsol=all)
 

\[ y \left (t \right ) = \left (t c_{4} +c_{2} \right ) {\mathrm e}^{3 t}+\cos \left (2 t \right ) c_{1} +c_{3} \sin \left (2 t \right )+3 t +2 \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 41

DSolve[y''''[t]-6*y'''[t]+13*y''[t]-24*y'[t]+36*y[t]==108*t,y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to 3 t+c_4 e^{3 t} t+c_3 e^{3 t}+c_1 \cos (2 t)+c_2 \sin (2 t)+2 \]