problem 27.1 (m)

Internal problem ID [13860]
Internal ﬁle name [OUTPUT/13032_Friday_February_23_2024_06_54_12_AM_54640770/index.tex]

Book: Ordinary Diﬀerential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section: Chapter 27. Diﬀerentiation and the Laplace transform. Additional Exercises. page 496
Problem number: 27.1 (m).
ODE order: 3.
ODE degree: 1.

\[ \boxed {y^{\prime \prime \prime }-27 y={\mathrm e}^{-3 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 3, y^{\prime \prime }\left (0\right ) = 4] \end {align*}

Solving using the Laplace transform method. Let \[ \mathcal {L}\left (y\right ) =Y(s) \] Taking the Laplace transform of the ode and using the relations that \begin {align*} \mathcal {L}\left (y^{\prime }\right )&= s Y(s) - y \left (0\right )\\ \mathcal {L}\left (y^{\prime \prime }\right ) &= s^2 Y(s) - y'(0) - s y \left (0\right )\\ \mathcal {L}\left (y^{\prime \prime \prime }\right ) &= s^3 Y(s) - y''(0) - s y'(0) - s^2 y \left (0\right ) \end {align*}

The given ode becomes an algebraic equation in the Laplace domain \[ s^{3} Y \left (s \right )-y^{\prime \prime }\left (0\right )-s y^{\prime }\left (0\right )-s^{2} y \left (0\right )-27 Y \left (s \right ) = \frac {1}{s +3}\tag {1} \] But the initial conditions are \begin {align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=3\\ y^{\prime \prime }\left (0\right )&=4 \end {align*}

Substituting these initial conditions in above in Eq (1) gives \[ s^{3} Y \left (s \right )-4-3 s -2 s^{2}-27 Y \left (s \right ) = \frac {1}{s +3} \] Solving the above equation for \(Y(s)\) results in \[ Y(s) = \frac {2 s^{3}+9 s^{2}+13 s +13}{\left (s +3\right ) \left (s^{3}-27\right )} \] Applying partial fractions decomposition results in \[ Y(s)= \frac {187}{162 \left (s -3\right )}-\frac {1}{54 \left (s +3\right )}+\frac {\frac {35}{81}-\frac {7 i \sqrt {3}}{81}}{s +\frac {3}{2}-\frac {3 i \sqrt {3}}{2}}+\frac {\frac {35}{81}+\frac {7 i \sqrt {3}}{81}}{s +\frac {3}{2}+\frac {3 i \sqrt {3}}{2}} \] The inverse Laplace of each term above is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (\frac {187}{162 \left (s -3\right )}\right ) &= \frac {187 \,{\mathrm e}^{3 t}}{162}\\ \mathcal {L}^{-1}\left (-\frac {1}{54 \left (s +3\right )}\right ) &= -\frac {{\mathrm e}^{-3 t}}{54}\\ \mathcal {L}^{-1}\left (\frac {\frac {35}{81}-\frac {7 i \sqrt {3}}{81}}{s +\frac {3}{2}-\frac {3 i \sqrt {3}}{2}}\right ) &= \frac {7 \left (5-i \sqrt {3}\right ) {\mathrm e}^{-\frac {3 \left (1-i \sqrt {3}\right ) t}{2}}}{81}\\ \mathcal {L}^{-1}\left (\frac {\frac {35}{81}+\frac {7 i \sqrt {3}}{81}}{s +\frac {3}{2}+\frac {3 i \sqrt {3}}{2}}\right ) &= \frac {7 \left (i \sqrt {3}+5\right ) {\mathrm e}^{-\frac {3 \left (1+i \sqrt {3}\right ) t}{2}}}{81} \end {align*}

Adding the above results and simplifying gives \[ y=\frac {92 \cosh \left (3 t \right )}{81}+\frac {95 \sinh \left (3 t \right )}{81}+\frac {14 \,{\mathrm e}^{-\frac {3 t}{2}} \left (\sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )+5 \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )\right )}{81} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {92 \cosh \left (3 t \right )}{81}+\frac {95 \sinh \left (3 t \right )}{81}+\frac {14 \,{\mathrm e}^{-\frac {3 t}{2}} \left (\sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )+5 \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )\right )}{81} \\ \end{align*}

Veriﬁcation of solutions

\[ y = \frac {92 \cosh \left (3 t \right )}{81}+\frac {95 \sinh \left (3 t \right )}{81}+\frac {14 \,{\mathrm e}^{-\frac {3 t}{2}} \left (\sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )+5 \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )\right )}{81} \] Veriﬁed OK.

18.13.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}y^{\prime \prime }-27 y={\mathrm e}^{-3 t}, y \left (0\right )=2, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=3, \left (\frac {d}{d t}y^{\prime }\right )\bigg | {\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=4\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d t}y^{\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 )=\frac {d}{d t}y^{\prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{3}^{\prime }\left (t \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{3}^{\prime }\left (t \right )={\mathrm e}^{-3 t}+27 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_{3}^{\prime }\left (t \right )={\mathrm e}^{-3 t}+27 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 ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (t \right )=\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 27 & 0 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (t \right )+\left [\begin {array}{c} 0 \\ 0 \\ {\mathrm e}^{-3 t} \end {array}\right ] \\ \bullet & {} & \textrm {Define the forcing function}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{f}}\left (t \right )=\left [\begin {array}{c} 0 \\ 0 \\ {\mathrm e}^{-3 t} \end {array}\right ] \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 27 & 0 & 0 \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}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]\right ], \left [-\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {3}{2}+\frac {3 \,\mathrm {I} \sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {3}{2}+\frac {3 \,\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {3}{2}+\frac {3 \,\mathrm {I} \sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [3, \left [\begin {array}{c} \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{3 t}\cdot \left [\begin {array}{c} \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}\right ) t}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{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 \\ {} & {} & {\mathrm e}^{-\frac {3 t}{2}}\cdot \left (\cos \left (\frac {3 \sqrt {3}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-\frac {3 t}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {3 \sqrt {3}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{\left (-\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {\cos \left (\frac {3 \sqrt {3}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{-\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}} \\ \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )-\mathrm {I} \sin \left (\frac {3 \sqrt {3}\, t}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{2}\left (t \right )={\mathrm e}^{-\frac {3 t}{2}}\cdot \left [\begin {array}{c} -\frac {\cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{18}-\frac {\sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{18} \\ -\frac {\cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{6}+\frac {\sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{6} \\ \cos \left (\frac {3 \sqrt {3}\, t}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (t \right )={\mathrm e}^{-\frac {3 t}{2}}\cdot \left [\begin {array}{c} -\frac {\sqrt {3}\, \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{18}+\frac {\sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{18} \\ \frac {\sqrt {3}\, \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{6}+\frac {\sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{6} \\ -\sin \left (\frac {3 \sqrt {3}\, t}{2}\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}+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (t \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\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}{ccc} \frac {{\mathrm e}^{3 t}}{9} & {\mathrm e}^{-\frac {3 t}{2}} \left (-\frac {\cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{18}-\frac {\sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{18}\right ) & {\mathrm e}^{-\frac {3 t}{2}} \left (-\frac {\sqrt {3}\, \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{18}+\frac {\sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{18}\right ) \\ \frac {{\mathrm e}^{3 t}}{3} & {\mathrm e}^{-\frac {3 t}{2}} \left (-\frac {\cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{6}+\frac {\sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{6}\right ) & {\mathrm e}^{-\frac {3 t}{2}} \left (\frac {\sqrt {3}\, \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{6}+\frac {\sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{6}\right ) \\ {\mathrm e}^{3 t} & {\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right ) & -{\mathrm e}^{-\frac {3 t}{2}} \sin \left (\frac {3 \sqrt {3}\, t}{2}\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 \phi \left (0\right )^{-1} \\ {} & \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}{ccc} \frac {{\mathrm e}^{3 t}}{9} & {\mathrm e}^{-\frac {3 t}{2}} \left (-\frac {\cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{18}-\frac {\sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{18}\right ) & {\mathrm e}^{-\frac {3 t}{2}} \left (-\frac {\sqrt {3}\, \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{18}+\frac {\sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{18}\right ) \\ \frac {{\mathrm e}^{3 t}}{3} & {\mathrm e}^{-\frac {3 t}{2}} \left (-\frac {\cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{6}+\frac {\sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{6}\right ) & {\mathrm e}^{-\frac {3 t}{2}} \left (\frac {\sqrt {3}\, \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{6}+\frac {\sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{6}\right ) \\ {\mathrm e}^{3 t} & {\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right ) & -{\mathrm e}^{-\frac {3 t}{2}} \sin \left (\frac {3 \sqrt {3}\, t}{2}\right ) \end {array}\right ]\cdot \left [\begin {array}{ccc} \frac {1}{9} & -\frac {1}{18} & -\frac {\sqrt {3}}{18} \\ \frac {1}{3} & -\frac {1}{6} & \frac {\sqrt {3}}{6} \\ 1 & 1 & 0 \end {array}\right ]^{-1} \\ {} & \circ & \textrm {Evaluate and simplify to get the fundamental matrix}\hspace {3pt} \\ {} & {} & \Phi \left (t \right )=\left [\begin {array}{ccc} \frac {{\mathrm e}^{3 t}}{3}+\frac {2 \,{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{3} & \frac {{\mathrm e}^{-\frac {3 t}{2}} \sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{9}-\frac {{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{9}+\frac {{\mathrm e}^{3 t}}{9} & \frac {{\mathrm e}^{3 t}}{27}-\frac {{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{27}-\frac {{\mathrm e}^{-\frac {3 t}{2}} \sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{27} \\ {\mathrm e}^{3 t}-{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )-{\mathrm e}^{-\frac {3 t}{2}} \sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right ) & \frac {{\mathrm e}^{3 t}}{3}+\frac {2 \,{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{3} & \frac {{\mathrm e}^{-\frac {3 t}{2}} \sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{9}-\frac {{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{9}+\frac {{\mathrm e}^{3 t}}{9} \\ 3 \,{\mathrm e}^{3 t}-3 \,{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )+3 \,{\mathrm e}^{-\frac {3 t}{2}} \sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right ) & {\mathrm e}^{3 t}-{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )-{\mathrm e}^{-\frac {3 t}{2}} \sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right ) & \frac {{\mathrm e}^{3 t}}{3}+\frac {2 \,{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{3} \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 )=\Phi \left (t \right )^{-1}\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}\Phi \left (s \right )^{-1}\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}\Phi \left (s \right )^{-1}\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} \frac {\left ({\mathrm e}^{6 t}-2 \,{\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {3 \sqrt {3}\, t}{2}\right ) \sqrt {3}+2 \,{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )-3\right ) {\mathrm e}^{-3 t}}{162} \\ \frac {\left ({\mathrm e}^{6 t}-4 \,{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )+3\right ) {\mathrm e}^{-3 t}}{54} \\ \frac {\left ({\mathrm e}^{6 t}+2 \,{\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {3 \sqrt {3}\, t}{2}\right ) \sqrt {3}+2 \,{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )-3\right ) {\mathrm e}^{-3 t}}{18} \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}+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (t \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (t \right )+\left [\begin {array}{c} \frac {\left ({\mathrm e}^{6 t}-2 \,{\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {3 \sqrt {3}\, t}{2}\right ) \sqrt {3}+2 \,{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )-3\right ) {\mathrm e}^{-3 t}}{162} \\ \frac {\left ({\mathrm e}^{6 t}-4 \,{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )+3\right ) {\mathrm e}^{-3 t}}{54} \\ \frac {\left ({\mathrm e}^{6 t}+2 \,{\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {3 \sqrt {3}\, t}{2}\right ) \sqrt {3}+2 \,{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )-3\right ) {\mathrm e}^{-3 t}}{18} \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=-\frac {{\mathrm e}^{-3 t} \left ({\mathrm e}^{\frac {3 t}{2}} \left (c_{3} \sqrt {3}+c_{2} -\frac {2}{9}\right ) \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )+{\mathrm e}^{\frac {3 t}{2}} \left (\left (c_{2} +\frac {2}{9}\right ) \sqrt {3}-c_{3} \right ) \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )+\frac {1}{3}+\left (-2 c_{1} -\frac {1}{9}\right ) {\mathrm e}^{6 t}\right )}{18} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y \left (0\right )=2 \\ {} & {} & 2=-\frac {c_{3} \sqrt {3}}{18}-\frac {c_{2}}{18}+\frac {c_{1}}{9} \\ \bullet & {} & \textrm {Calculate the 1st derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{-3 t} \left ({\mathrm e}^{\frac {3 t}{2}} \left (c_{3} \sqrt {3}+c_{2} -\frac {2}{9}\right ) \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )+{\mathrm e}^{\frac {3 t}{2}} \left (\left (c_{2} +\frac {2}{9}\right ) \sqrt {3}-c_{3} \right ) \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )+\frac {1}{3}+\left (-2 c_{1} -\frac {1}{9}\right ) {\mathrm e}^{6 t}\right )}{6}-\frac {{\mathrm e}^{-3 t} \left (\frac {3 \,{\mathrm e}^{\frac {3 t}{2}} \left (c_{3} \sqrt {3}+c_{2} -\frac {2}{9}\right ) \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{2}-\frac {3 \,{\mathrm e}^{\frac {3 t}{2}} \left (c_{3} \sqrt {3}+c_{2} -\frac {2}{9}\right ) \sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{2}+\frac {3 \,{\mathrm e}^{\frac {3 t}{2}} \left (\left (c_{2} +\frac {2}{9}\right ) \sqrt {3}-c_{3} \right ) \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{2}+\frac {3 \,{\mathrm e}^{\frac {3 t}{2}} \left (\left (c_{2} +\frac {2}{9}\right ) \sqrt {3}-c_{3} \right ) \sqrt {3}\, \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{2}+6 \left (-2 c_{1} -\frac {1}{9}\right ) {\mathrm e}^{6 t}\right )}{18} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=3 \\ {} & {} & 3=\frac {c_{3} \sqrt {3}}{12}+\frac {c_{2}}{12}+\frac {c_{1}}{3}+\frac {1}{18}-\frac {\left (\left (c_{2} +\frac {2}{9}\right ) \sqrt {3}-c_{3} \right ) \sqrt {3}}{12} \\ \bullet & {} & \textrm {Calculate the 2nd derivative of the solution}\hspace {3pt} \\ {} & {} & \frac {d}{d t}y^{\prime }=-\frac {{\mathrm e}^{-3 t} \left ({\mathrm e}^{\frac {3 t}{2}} \left (c_{3} \sqrt {3}+c_{2} -\frac {2}{9}\right ) \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )+{\mathrm e}^{\frac {3 t}{2}} \left (\left (c_{2} +\frac {2}{9}\right ) \sqrt {3}-c_{3} \right ) \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )+\frac {1}{3}+\left (-2 c_{1} -\frac {1}{9}\right ) {\mathrm e}^{6 t}\right )}{2}+\frac {{\mathrm e}^{-3 t} \left (\frac {3 \,{\mathrm e}^{\frac {3 t}{2}} \left (c_{3} \sqrt {3}+c_{2} -\frac {2}{9}\right ) \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{2}-\frac {3 \,{\mathrm e}^{\frac {3 t}{2}} \left (c_{3} \sqrt {3}+c_{2} -\frac {2}{9}\right ) \sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{2}+\frac {3 \,{\mathrm e}^{\frac {3 t}{2}} \left (\left (c_{2} +\frac {2}{9}\right ) \sqrt {3}-c_{3} \right ) \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{2}+\frac {3 \,{\mathrm e}^{\frac {3 t}{2}} \left (\left (c_{2} +\frac {2}{9}\right ) \sqrt {3}-c_{3} \right ) \sqrt {3}\, \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{2}+6 \left (-2 c_{1} -\frac {1}{9}\right ) {\mathrm e}^{6 t}\right )}{3}-\frac {{\mathrm e}^{-3 t} \left (-\frac {9 \,{\mathrm e}^{\frac {3 t}{2}} \left (c_{3} \sqrt {3}+c_{2} -\frac {2}{9}\right ) \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{2}-\frac {9 \,{\mathrm e}^{\frac {3 t}{2}} \left (c_{3} \sqrt {3}+c_{2} -\frac {2}{9}\right ) \sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{2}-\frac {9 \,{\mathrm e}^{\frac {3 t}{2}} \left (\left (c_{2} +\frac {2}{9}\right ) \sqrt {3}-c_{3} \right ) \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{2}+\frac {9 \,{\mathrm e}^{\frac {3 t}{2}} \left (\left (c_{2} +\frac {2}{9}\right ) \sqrt {3}-c_{3} \right ) \sqrt {3}\, \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{2}+36 \left (-2 c_{1} -\frac {1}{9}\right ) {\mathrm e}^{6 t}\right )}{18} \\ \bullet & {} & \textrm {Use the initial condition}\hspace {3pt} \left (\frac {d}{d t}y^{\prime }\right )\bigg | {\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=4 \\ {} & {} & 4=\frac {c_{3} \sqrt {3}}{4}+\frac {c_{2}}{4}+c_{1} -\frac {1}{6}+\frac {\left (\left (c_{2} +\frac {2}{9}\right ) \sqrt {3}-c_{3} \right ) \sqrt {3}}{4} \\ \bullet & {} & \textrm {Solve for the unknown coefficients}\hspace {3pt} \\ {} & {} & \left \{c_{1} =\frac {31}{3}, c_{2} =-\frac {19}{3}, c_{3} =-3 \sqrt {3}\right \} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {{\mathrm e}^{-3 t} \left (28 \,{\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {3 \sqrt {3}\, t}{2}\right ) \sqrt {3}+140 \,{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )+187 \,{\mathrm e}^{6 t}-3\right )}{162} \end {array} \]

Maple trace

`Methods for third order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 3; linear nonhomogeneous with symmetry [0,1] 
trying high order linear exact nonhomogeneous 
trying differential order: 3; missing the dependent variable 
checking if the LODE has constant coefficients 
<- constant coefficients successful`

\[ y \left (t \right ) = \frac {14 \sqrt {3}\, {\mathrm e}^{-\frac {3 t}{2}} \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{81}+\frac {70 \,{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{81}+\frac {92 \cosh \left (3 t \right )}{81}+\frac {95 \sinh \left (3 t \right )}{81} \]

\[ y(t)\to \frac {1}{162} e^{-3 t} \left (187 e^{6 t}+28 \sqrt {3} e^{3 t/2} \sin \left (\frac {3 \sqrt {3} t}{2}\right )+140 e^{3 t/2} \cos \left (\frac {3 \sqrt {3} t}{2}\right )-3\right ) \]

18.13 problem 27.1 (m)

18.13.1 Maple step by step solution