21.3 problem 3

Internal problem ID [13236]
Internal file name [OUTPUT/11892_Tuesday_December_05_2023_12_12_49_PM_88058161/index.tex]

Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 6. Laplace transform. Section 6.6. page 624
Problem number: 3.
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "second_order_laplace", "second_order_linear_constant_coeff"

Maple gives the following as the ode type

[[_2nd_order, _linear, _nonhomogeneous]]

\[ \boxed {y^{\prime \prime }+y^{\prime }+8 y=\left (1-\operatorname {Heaviside}\left (t -4\right )\right ) \cos \left (t -4\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}

21.3.1 Existence and uniqueness analysis

This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime \prime } + p(t)y^{\prime } + q(t) y &= F \end {align*}

Where here \begin {align*} p(t) &=1\\ q(t) &=8\\ F &=\left (1-\operatorname {Heaviside}\left (t -4\right )\right ) \cos \left (t -4\right ) \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }+y^{\prime }+8 y = \left (1-\operatorname {Heaviside}\left (t -4\right )\right ) \cos \left (t -4\right ) \end {align*}

The domain of \(p(t)=1\) is \[ \{-\infty

Solving using the Laplace transform method. Let \begin {align*} \mathcal {L}\left (y\right ) =Y(s) \end {align*}

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 ) \end {align*}

The given ode now becomes an algebraic equation in the Laplace domain \begin {align*} s^{2} Y \left (s \right )-y^{\prime }\left (0\right )-s y \left (0\right )+s Y \left (s \right )-y \left (0\right )+8 Y \left (s \right ) = -\frac {-\sin \left (4\right )+s \left ({\mathrm e}^{-4 s}-\cos \left (4\right )\right )}{s^{2}+1}\tag {1} \end {align*}

But the initial conditions are \begin {align*} y \left (0\right )&=0\\ y'(0) &=0 \end {align*}

Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )+s Y \left (s \right )+8 Y \left (s \right ) = -\frac {-\sin \left (4\right )+s \left ({\mathrm e}^{-4 s}-\cos \left (4\right )\right )}{s^{2}+1} \end {align*}

Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {s \cos \left (4\right )-s \,{\mathrm e}^{-4 s}+\sin \left (4\right )}{\left (s^{2}+1\right ) \left (s^{2}+s +8\right )} \end {align*}

Taking the inverse Laplace transform gives \begin {align*} y&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (\frac {s \cos \left (4\right )-s \,{\mathrm e}^{-4 s}+\sin \left (4\right )}{\left (s^{2}+1\right ) \left (s^{2}+s +8\right )}\right )\\ &= \frac {\cos \left (t \right ) \left (7 \cos \left (4\right )-\sin \left (4\right )\right )}{50}+\frac {\left (-31 \cos \left (\frac {\sqrt {31}\, t}{2}\right ) \left (7 \cos \left (4\right )-\sin \left (4\right )\right )+\sin \left (\frac {\sqrt {31}\, t}{2}\right ) \sqrt {31}\, \left (-9 \cos \left (4\right )-13 \sin \left (4\right )\right )\right ) {\mathrm e}^{-\frac {t}{2}}}{1550}-\frac {\sin \left (t \right ) \left (-\cos \left (4\right )-7 \sin \left (4\right )\right )}{50}+\frac {\left (-31 \sin \left (t -4\right )-217 \cos \left (t -4\right )+{\mathrm e}^{-\frac {t}{2}+2} \left (9 \sqrt {31}\, \sin \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )+217 \cos \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )\right )\right ) \operatorname {Heaviside}\left (t -4\right )}{1550} \end {align*}

Converting the above solution to piecewise it becomes \[ y = \left \{\begin {array}{cc} \frac {\cos \left (t \right ) \left (7 \cos \left (4\right )-\sin \left (4\right )\right )}{50}+\frac {\left (-31 \cos \left (\frac {\sqrt {31}\, t}{2}\right ) \left (7 \cos \left (4\right )-\sin \left (4\right )\right )+\sin \left (\frac {\sqrt {31}\, t}{2}\right ) \sqrt {31}\, \left (-9 \cos \left (4\right )-13 \sin \left (4\right )\right )\right ) {\mathrm e}^{-\frac {t}{2}}}{1550}-\frac {\sin \left (t \right ) \left (-\cos \left (4\right )-7 \sin \left (4\right )\right )}{50} & t <4 \\ \frac {\cos \left (t \right ) \left (7 \cos \left (4\right )-\sin \left (4\right )\right )}{50}+\frac {\left (-31 \cos \left (\frac {\sqrt {31}\, t}{2}\right ) \left (7 \cos \left (4\right )-\sin \left (4\right )\right )+\sin \left (\frac {\sqrt {31}\, t}{2}\right ) \sqrt {31}\, \left (-9 \cos \left (4\right )-13 \sin \left (4\right )\right )\right ) {\mathrm e}^{-\frac {t}{2}}}{1550}-\frac {\sin \left (t \right ) \left (-\cos \left (4\right )-7 \sin \left (4\right )\right )}{50}-\frac {\sin \left (t -4\right )}{50}-\frac {7 \cos \left (t -4\right )}{50}+\frac {{\mathrm e}^{-\frac {t}{2}+2} \left (9 \sqrt {31}\, \sin \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )+217 \cos \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )\right )}{1550} & 4\le t \end {array}\right . \] Simplifying the solution gives \[ y = \frac {\left (\left \{\begin {array}{cc} \left (7 \cos \left (t \right )+\sin \left (t \right )\right ) \cos \left (4\right )-\sin \left (4\right ) \left (\cos \left (t \right )-7 \sin \left (t \right )\right ) & t <4 \\ \frac {{\mathrm e}^{-\frac {t}{2}+2} \left (9 \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \sqrt {31}\, \cos \left (2 \sqrt {31}\right )-9 \cos \left (\frac {\sqrt {31}\, t}{2}\right ) \sqrt {31}\, \sin \left (2 \sqrt {31}\right )+217 \cos \left (\frac {\sqrt {31}\, t}{2}\right ) \cos \left (2 \sqrt {31}\right )+217 \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \sin \left (2 \sqrt {31}\right )\right )}{31} & 4\le t \end {array}\right .\right )}{50}+\frac {\left (-7 \cos \left (4\right )+\sin \left (4\right )\right ) {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{50}+\frac {\left (-9 \cos \left (4\right )-13 \sin \left (4\right )\right ) \sqrt {31}\, {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{1550} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (\left \{\begin {array}{cc} \left (7 \cos \left (t \right )+\sin \left (t \right )\right ) \cos \left (4\right )-\sin \left (4\right ) \left (\cos \left (t \right )-7 \sin \left (t \right )\right ) & t <4 \\ \frac {{\mathrm e}^{-\frac {t}{2}+2} \left (9 \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \sqrt {31}\, \cos \left (2 \sqrt {31}\right )-9 \cos \left (\frac {\sqrt {31}\, t}{2}\right ) \sqrt {31}\, \sin \left (2 \sqrt {31}\right )+217 \cos \left (\frac {\sqrt {31}\, t}{2}\right ) \cos \left (2 \sqrt {31}\right )+217 \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \sin \left (2 \sqrt {31}\right )\right )}{31} & 4\le t \end {array}\right .\right )}{50}+\frac {\left (-7 \cos \left (4\right )+\sin \left (4\right )\right ) {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{50}+\frac {\left (-9 \cos \left (4\right )-13 \sin \left (4\right )\right ) \sqrt {31}\, {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{1550} \\ \end{align*}

Verification of solutions

\[ y = \frac {\left (\left \{\begin {array}{cc} \left (7 \cos \left (t \right )+\sin \left (t \right )\right ) \cos \left (4\right )-\sin \left (4\right ) \left (\cos \left (t \right )-7 \sin \left (t \right )\right ) & t <4 \\ \frac {{\mathrm e}^{-\frac {t}{2}+2} \left (9 \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \sqrt {31}\, \cos \left (2 \sqrt {31}\right )-9 \cos \left (\frac {\sqrt {31}\, t}{2}\right ) \sqrt {31}\, \sin \left (2 \sqrt {31}\right )+217 \cos \left (\frac {\sqrt {31}\, t}{2}\right ) \cos \left (2 \sqrt {31}\right )+217 \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \sin \left (2 \sqrt {31}\right )\right )}{31} & 4\le t \end {array}\right .\right )}{50}+\frac {\left (-7 \cos \left (4\right )+\sin \left (4\right )\right ) {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{50}+\frac {\left (-9 \cos \left (4\right )-13 \sin \left (4\right )\right ) \sqrt {31}\, {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{1550} \] Verified OK.

21.3.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime \prime }+y^{\prime }+8 y=\left (1-\mathit {Heaviside}\left (t -4\right )\right ) \cos \left (t -4\right ), y \left (0\right )=0, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=-8 y-\cos \left (t -4\right ) \mathit {Heaviside}\left (t -4\right )+\cos \left (t -4\right )-y^{\prime } \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & y^{\prime \prime }+y^{\prime }+8 y=-\left (-1+\mathit {Heaviside}\left (t -4\right )\right ) \cos \left (t -4\right ) \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+r +8=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {\left (-1\right )\pm \left (\sqrt {-31}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {31}}{2}, -\frac {1}{2}+\frac {\mathrm {I} \sqrt {31}}{2}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )={\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right ) \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )={\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y=c_{1} y_{1}\left (t \right )+c_{2} y_{2}\left (t \right )+y_{p}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right )+c_{2} {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right )+y_{p}\left (t \right ) \\ \square & {} & \textrm {Find a particular solution}\hspace {3pt} y_{p}\left (t \right )\hspace {3pt}\textrm {of the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Use variation of parameters to find}\hspace {3pt} y_{p}\hspace {3pt}\textrm {here}\hspace {3pt} f \left (t \right )\hspace {3pt}\textrm {is the forcing function}\hspace {3pt} \\ {} & {} & \left [y_{p}\left (t \right )=-y_{1}\left (t \right ) \left (\int \frac {y_{2}\left (t \right ) f \left (t \right )}{W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )}d t \right )+y_{2}\left (t \right ) \left (\int \frac {y_{1}\left (t \right ) f \left (t \right )}{W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )}d t \right ), f \left (t \right )=-\left (-1+\mathit {Heaviside}\left (t -4\right )\right ) \cos \left (t -4\right )\right ] \\ {} & \circ & \textrm {Wronskian of solutions of the homogeneous equation}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )=\left [\begin {array}{cc} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right ) & {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \\ -\frac {{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{2}-\frac {{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \sqrt {31}}{2} & -\frac {{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{2}+\frac {{\mathrm e}^{-\frac {t}{2}} \sqrt {31}\, \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{2} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )=\frac {\sqrt {31}\, {\mathrm e}^{-t}}{2} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (t \right ) \\ {} & {} & y_{p}\left (t \right )=-\frac {2 \sqrt {31}\, {\mathrm e}^{-\frac {t}{2}} \left (\sin \left (\frac {\sqrt {31}\, t}{2}\right ) \left (\int {\mathrm e}^{\frac {t}{2}} \cos \left (t -4\right ) \cos \left (\frac {\sqrt {31}\, t}{2}\right ) \left (-1+\mathit {Heaviside}\left (t -4\right )\right )d t \right )-\cos \left (\frac {\sqrt {31}\, t}{2}\right ) \left (\int {\mathrm e}^{\frac {t}{2}} \cos \left (t -4\right ) \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \left (-1+\mathit {Heaviside}\left (t -4\right )\right )d t \right )\right )}{31} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (t \right )=\frac {7 \,{\mathrm e}^{-\frac {t}{2}+2} \mathit {Heaviside}\left (t -4\right ) \cos \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )}{50}+\frac {9 \,{\mathrm e}^{-\frac {t}{2}+2} \sqrt {31}\, \mathit {Heaviside}\left (t -4\right ) \sin \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )}{1550}-\frac {7 \left (-1+\mathit {Heaviside}\left (t -4\right )\right ) \left (\cos \left (t -4\right )+\frac {\sin \left (t -4\right )}{7}\right )}{50} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right )+c_{2} {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right )+\frac {7 \,{\mathrm e}^{-\frac {t}{2}+2} \mathit {Heaviside}\left (t -4\right ) \cos \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )}{50}+\frac {9 \,{\mathrm e}^{-\frac {t}{2}+2} \sqrt {31}\, \mathit {Heaviside}\left (t -4\right ) \sin \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )}{1550}-\frac {7 \left (-1+\mathit {Heaviside}\left (t -4\right )\right ) \left (\cos \left (t -4\right )+\frac {\sin \left (t -4\right )}{7}\right )}{50} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right )+c_{2} {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right )+\frac {7 {\mathrm e}^{-\frac {t}{2}+2} \mathit {Heaviside}\left (t -4\right ) \cos \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )}{50}+\frac {9 {\mathrm e}^{-\frac {t}{2}+2} \sqrt {31}\, \mathit {Heaviside}\left (t -4\right ) \sin \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )}{1550}-\frac {7 \left (-1+\mathit {Heaviside}\left (t -4\right )\right ) \left (\cos \left (t -4\right )+\frac {\sin \left (t -4\right )}{7}\right )}{50} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=c_{1} +\frac {7 \cos \left (4\right )}{50}-\frac {\sin \left (4\right )}{50} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {c_{1} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{2}-\frac {c_{1} {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \sqrt {31}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{2}+\frac {c_{2} {\mathrm e}^{-\frac {t}{2}} \sqrt {31}\, \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{2}+\frac {{\mathrm e}^{-\frac {t}{2}+2} \mathit {Heaviside}\left (t -4\right ) \cos \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )}{50}+\frac {7 \,{\mathrm e}^{-\frac {t}{2}+2} \mathit {Dirac}\left (t -4\right ) \cos \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )}{50}-\frac {113 \,{\mathrm e}^{-\frac {t}{2}+2} \sqrt {31}\, \mathit {Heaviside}\left (t -4\right ) \sin \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )}{1550}+\frac {9 \,{\mathrm e}^{-\frac {t}{2}+2} \sqrt {31}\, \mathit {Dirac}\left (t -4\right ) \sin \left (\frac {\sqrt {31}\, \left (t -4\right )}{2}\right )}{1550}-\frac {7 \mathit {Dirac}\left (t -4\right ) \left (\cos \left (t -4\right )+\frac {\sin \left (t -4\right )}{7}\right )}{50}-\frac {7 \left (-1+\mathit {Heaviside}\left (t -4\right )\right ) \left (-\sin \left (t -4\right )+\frac {\cos \left (t -4\right )}{7}\right )}{50} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0 \\ {} & {} & 0=-\frac {c_{1}}{2}+\frac {c_{2} \sqrt {31}}{2}+\frac {7 \sin \left (4\right )}{50}+\frac {\cos \left (4\right )}{50} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =-\frac {7 \cos \left (4\right )}{50}+\frac {\sin \left (4\right )}{50}, c_{2} =-\frac {\sqrt {31}\, \left (9 \cos \left (4\right )+13 \sin \left (4\right )\right )}{1550}\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=-\frac {9 \mathit {Heaviside}\left (t -4\right ) \left (\left (\sqrt {31}\, \sin \left (2 \sqrt {31}\right )-\frac {217 \cos \left (2 \sqrt {31}\right )}{9}\right ) \cos \left (\frac {\sqrt {31}\, t}{2}\right )-\frac {217 \left (\frac {9 \sqrt {31}\, \cos \left (2 \sqrt {31}\right )}{217}+\sin \left (2 \sqrt {31}\right )\right ) \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{9}\right ) {\mathrm e}^{-\frac {t}{2}+2}}{1550}-\frac {7 \left (\cos \left (4\right )-\frac {\sin \left (4\right )}{7}\right ) {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{50}-\frac {9 \,{\mathrm e}^{-\frac {t}{2}} \left (\cos \left (4\right )+\frac {13 \sin \left (4\right )}{9}\right ) \sqrt {31}\, \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{1550}-\frac {7 \left (\left (\cos \left (t \right )+\frac {\sin \left (t \right )}{7}\right ) \cos \left (4\right )-\frac {\sin \left (4\right ) \left (\cos \left (t \right )-7 \sin \left (t \right )\right )}{7}\right ) \left (-1+\mathit {Heaviside}\left (t -4\right )\right )}{50} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=-\frac {9 \mathit {Heaviside}\left (t -4\right ) \left (\left (\sqrt {31}\, \sin \left (2 \sqrt {31}\right )-\frac {217 \cos \left (2 \sqrt {31}\right )}{9}\right ) \cos \left (\frac {\sqrt {31}\, t}{2}\right )-\frac {217 \left (\frac {9 \sqrt {31}\, \cos \left (2 \sqrt {31}\right )}{217}+\sin \left (2 \sqrt {31}\right )\right ) \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{9}\right ) {\mathrm e}^{-\frac {t}{2}+2}}{1550}-\frac {7 \left (\cos \left (4\right )-\frac {\sin \left (4\right )}{7}\right ) {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{50}-\frac {9 \,{\mathrm e}^{-\frac {t}{2}} \left (\cos \left (4\right )+\frac {13 \sin \left (4\right )}{9}\right ) \sqrt {31}\, \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{1550}-\frac {7 \left (\left (\cos \left (t \right )+\frac {\sin \left (t \right )}{7}\right ) \cos \left (4\right )-\frac {\sin \left (4\right ) \left (\cos \left (t \right )-7 \sin \left (t \right )\right )}{7}\right ) \left (-1+\mathit {Heaviside}\left (t -4\right )\right )}{50} \end {array} \]

Maple trace

`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`
 

✓ Solution by Maple

Time used: 6.797 (sec). Leaf size: 128

dsolve([diff(y(t),t$2)+diff(y(t),t)+8*y(t)=(1-Heaviside(t-4))*cos(t-4),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y \left (t \right ) = -\frac {9 \operatorname {Heaviside}\left (t -4\right ) \left (\left (\sin \left (2 \sqrt {31}\right ) \sqrt {31}-\frac {217 \cos \left (2 \sqrt {31}\right )}{9}\right ) \cos \left (\frac {\sqrt {31}\, t}{2}\right )-\frac {217 \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \left (\frac {9 \sqrt {31}\, \cos \left (2 \sqrt {31}\right )}{217}+\sin \left (2 \sqrt {31}\right )\right )}{9}\right ) {\mathrm e}^{-\frac {t}{2}+2}}{1550}-\frac {7 \,{\mathrm e}^{-\frac {t}{2}} \left (\cos \left (4\right )-\frac {\sin \left (4\right )}{7}\right ) \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{50}-\frac {9 \left (\cos \left (4\right )+\frac {13 \sin \left (4\right )}{9}\right ) \sqrt {31}\, {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{1550}-\frac {7 \left (\left (\cos \left (t \right )+\frac {\sin \left (t \right )}{7}\right ) \cos \left (4\right )-\frac {\sin \left (4\right ) \left (-7 \sin \left (t \right )+\cos \left (t \right )\right )}{7}\right ) \left (-1+\operatorname {Heaviside}\left (t -4\right )\right )}{50} \]

✓ Solution by Mathematica

Time used: 4.688 (sec). Leaf size: 207

DSolve[{y''[t]+y'[t]+8*y[t]==(1-UnitStep[t-4])*Cos[t-4],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {e^{-t/2} \left (\theta (4-t) \left (-31 e^{t/2} \sin (4-t)-9 \sqrt {31} e^2 \sin \left (\frac {1}{2} \sqrt {31} (t-4)\right )+217 e^{t/2} \cos (4-t)-217 e^2 \cos \left (\frac {1}{2} \sqrt {31} (t-4)\right )\right )+9 \sqrt {31} e^2 \sin \left (\frac {1}{2} \sqrt {31} (t-4)\right )-13 \sqrt {31} \sin (4) \sin \left (\frac {\sqrt {31} t}{2}\right )+217 e^2 \cos \left (\frac {1}{2} \sqrt {31} (t-4)\right )-217 \cos (4) \cos \left (\frac {\sqrt {31} t}{2}\right )-9 \sqrt {31} \cos (4) \sin \left (\frac {\sqrt {31} t}{2}\right )+31 \sin (4) \cos \left (\frac {\sqrt {31} t}{2}\right )\right )}{1550} \]