21.4 problem 4

Internal problem ID [13237]
Internal file name [OUTPUT/11893_Tuesday_December_05_2023_12_12_50_PM_50415031/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: 4.
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]]

\[ \boxed {y^{\prime \prime }+y^{\prime }+3 y=\left (1-\operatorname {Heaviside}\left (t -2\right )\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}} \sin \left (t -2\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align*}

21.4.1 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) &=3\\ F &=-\left (-1+\operatorname {Heaviside}\left (t -2\right )\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}} \sin \left (t -2\right ) \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }+y^{\prime }+3 y = -\left (-1+\operatorname {Heaviside}\left (t -2\right )\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}} \sin \left (t -2\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 )+3 Y \left (s \right ) = 5 i \left (\frac {-{\mathrm e}^{\frac {1}{5}-2 i}+{\mathrm e}^{-2 s}}{10 s +1-10 i}+\frac {{\mathrm e}^{\frac {1}{5}+2 i}-{\mathrm e}^{-2 s}}{10 s +1+10 i}\right )\tag {1} \end {align*}

But the initial conditions are \begin {align*} y \left (0\right )&=1\\ y'(0) &=2 \end {align*}

Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )-3-s +s Y \left (s \right )+3 Y \left (s \right ) = 5 i \left (\frac {-{\mathrm e}^{\frac {1}{5}-2 i}+{\mathrm e}^{-2 s}}{10 s +1-10 i}+\frac {{\mathrm e}^{\frac {1}{5}+2 i}-{\mathrm e}^{-2 s}}{10 s +1+10 i}\right ) \end {align*}

Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {50 i {\mathrm e}^{\frac {1}{5}-2 i} s -50 i {\mathrm e}^{\frac {1}{5}+2 i} s -100 s^{3}+5 i {\mathrm e}^{\frac {1}{5}-2 i}-5 i {\mathrm e}^{\frac {1}{5}+2 i}-320 s^{2}+100 \,{\mathrm e}^{-2 s}-50 \,{\mathrm e}^{\frac {1}{5}-2 i}-50 \,{\mathrm e}^{\frac {1}{5}+2 i}-161 s -303}{\left (-10 s -1+10 i\right ) \left (10 s +1+10 i\right ) \left (s^{2}+s +3\right )} \end {align*}

Taking the inverse Laplace transform gives \begin {align*} y&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (\frac {50 i {\mathrm e}^{\frac {1}{5}-2 i} s -50 i {\mathrm e}^{\frac {1}{5}+2 i} s -100 s^{3}+5 i {\mathrm e}^{\frac {1}{5}-2 i}-5 i {\mathrm e}^{\frac {1}{5}+2 i}-320 s^{2}+100 \,{\mathrm e}^{-2 s}-50 \,{\mathrm e}^{\frac {1}{5}-2 i}-50 \,{\mathrm e}^{\frac {1}{5}+2 i}-161 s -303}{\left (-10 s -1+10 i\right ) \left (10 s +1+10 i\right ) \left (s^{2}+s +3\right )}\right )\\ &= \left (\frac {80}{1838780161}-\frac {191 i}{1838780161}\right ) \left (2144050 \,{\mathrm e}^{\frac {1}{5}+2 i-\frac {t}{2}}+\left (3430480+8190271 i\right ) {\mathrm e}^{-\frac {t}{2}}+\left (-1504050+1528000 i\right ) {\mathrm e}^{\frac {1}{5}-2 i-\frac {t}{2}}\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (-\frac {4000}{42881}+\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}-i\right ) \left (t -2\right )}+\left (-\frac {4000}{42881}-\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}+i\right ) \left (t -2\right )}+\left (\frac {3975}{586570871359}+\frac {3910 i}{586570871359}\right ) \sqrt {11}\, \left (-4974196 \,{\mathrm e}^{\frac {1}{5}+2 i-\frac {t}{2}}+\left (34090395-33532942 i\right ) {\mathrm e}^{-\frac {t}{2}}+\left (-82004+4973520 i\right ) {\mathrm e}^{\frac {1}{5}-2 i-\frac {t}{2}}\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right )+\frac {100 \left (2 \,{\mathrm e}^{-\frac {t}{2}+1} \left (159 \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right )-440 \cos \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right )\right )+11 \left (-191 \sin \left (t -2\right )+80 \cos \left (t -2\right )\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}}\right ) \operatorname {Heaviside}\left (t -2\right )}{471691} \end {align*}

Converting the above solution to piecewise it becomes \[ y = \left \{\begin {array}{cc} \left (\frac {80}{1838780161}-\frac {191 i}{1838780161}\right ) \left (2144050 \,{\mathrm e}^{\frac {1}{5}+2 i-\frac {t}{2}}+\left (3430480+8190271 i\right ) {\mathrm e}^{-\frac {t}{2}}+\left (-1504050+1528000 i\right ) {\mathrm e}^{\frac {1}{5}-2 i-\frac {t}{2}}\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (-\frac {4000}{42881}+\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}-i\right ) \left (t -2\right )}+\left (-\frac {4000}{42881}-\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}+i\right ) \left (t -2\right )}+\left (\frac {3975}{586570871359}+\frac {3910 i}{586570871359}\right ) \sqrt {11}\, \left (-4974196 \,{\mathrm e}^{\frac {1}{5}+2 i-\frac {t}{2}}+\left (34090395-33532942 i\right ) {\mathrm e}^{-\frac {t}{2}}+\left (-82004+4973520 i\right ) {\mathrm e}^{\frac {1}{5}-2 i-\frac {t}{2}}\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right ) & t <2 \\ \left (\frac {80}{1838780161}-\frac {191 i}{1838780161}\right ) \left (2144050 \,{\mathrm e}^{\frac {1}{5}+2 i-\frac {t}{2}}+\left (3430480+8190271 i\right ) {\mathrm e}^{-\frac {t}{2}}+\left (-1504050+1528000 i\right ) {\mathrm e}^{\frac {1}{5}-2 i-\frac {t}{2}}\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (-\frac {4000}{42881}+\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}-i\right ) \left (t -2\right )}+\left (-\frac {4000}{42881}-\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}+i\right ) \left (t -2\right )}+\left (\frac {3975}{586570871359}+\frac {3910 i}{586570871359}\right ) \sqrt {11}\, \left (-4974196 \,{\mathrm e}^{\frac {1}{5}+2 i-\frac {t}{2}}+\left (34090395-33532942 i\right ) {\mathrm e}^{-\frac {t}{2}}+\left (-82004+4973520 i\right ) {\mathrm e}^{\frac {1}{5}-2 i-\frac {t}{2}}\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right )+\frac {200 \,{\mathrm e}^{-\frac {t}{2}+1} \left (159 \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right )-440 \cos \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right )\right )}{471691}+\frac {100 \left (-191 \sin \left (t -2\right )+80 \cos \left (t -2\right )\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}}}{42881} & 2\le t \end {array}\right . \] Simplifying the solution gives \[ y = \frac {50 \left (\left \{\begin {array}{cc} \left (-80+191 i\right ) {\mathrm e}^{\left (-\frac {1}{10}-i\right ) \left (t -2\right )}+\left (-80-191 i\right ) {\mathrm e}^{\left (-\frac {1}{10}+i\right ) \left (t -2\right )} & t <2 \\ -\frac {4 \,{\mathrm e}^{-\frac {t}{2}+1} \left (159 \cos \left (\frac {\sqrt {11}\, t}{2}\right ) \sin \left (\sqrt {11}\right ) \sqrt {11}-159 \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \cos \left (\sqrt {11}\right ) \sqrt {11}+440 \cos \left (\frac {\sqrt {11}\, t}{2}\right ) \cos \left (\sqrt {11}\right )+440 \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \sin \left (\sqrt {11}\right )\right )}{11} & 2\le t \end {array}\right .\right )}{42881}+\frac {\left (\left (88000 \cos \left (2\right )+210100 \sin \left (2\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (-31800 \cos \left (2\right )+31280 \sin \left (2\right )\right ) \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )\right ) {\mathrm e}^{\frac {1}{5}-\frac {t}{2}}}{471691}+\frac {5 \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{11}+\cos \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {50 \left (\left \{\begin {array}{cc} \left (-80+191 i\right ) {\mathrm e}^{\left (-\frac {1}{10}-i\right ) \left (t -2\right )}+\left (-80-191 i\right ) {\mathrm e}^{\left (-\frac {1}{10}+i\right ) \left (t -2\right )} & t <2 \\ -\frac {4 \,{\mathrm e}^{-\frac {t}{2}+1} \left (159 \cos \left (\frac {\sqrt {11}\, t}{2}\right ) \sin \left (\sqrt {11}\right ) \sqrt {11}-159 \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \cos \left (\sqrt {11}\right ) \sqrt {11}+440 \cos \left (\frac {\sqrt {11}\, t}{2}\right ) \cos \left (\sqrt {11}\right )+440 \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \sin \left (\sqrt {11}\right )\right )}{11} & 2\le t \end {array}\right .\right )}{42881}+\frac {\left (\left (88000 \cos \left (2\right )+210100 \sin \left (2\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (-31800 \cos \left (2\right )+31280 \sin \left (2\right )\right ) \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )\right ) {\mathrm e}^{\frac {1}{5}-\frac {t}{2}}}{471691}+\frac {5 \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{11}+\cos \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \\ \end{align*}

Verification of solutions

\[ y = \frac {50 \left (\left \{\begin {array}{cc} \left (-80+191 i\right ) {\mathrm e}^{\left (-\frac {1}{10}-i\right ) \left (t -2\right )}+\left (-80-191 i\right ) {\mathrm e}^{\left (-\frac {1}{10}+i\right ) \left (t -2\right )} & t <2 \\ -\frac {4 \,{\mathrm e}^{-\frac {t}{2}+1} \left (159 \cos \left (\frac {\sqrt {11}\, t}{2}\right ) \sin \left (\sqrt {11}\right ) \sqrt {11}-159 \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \cos \left (\sqrt {11}\right ) \sqrt {11}+440 \cos \left (\frac {\sqrt {11}\, t}{2}\right ) \cos \left (\sqrt {11}\right )+440 \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \sin \left (\sqrt {11}\right )\right )}{11} & 2\le t \end {array}\right .\right )}{42881}+\frac {\left (\left (88000 \cos \left (2\right )+210100 \sin \left (2\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (-31800 \cos \left (2\right )+31280 \sin \left (2\right )\right ) \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )\right ) {\mathrm e}^{\frac {1}{5}-\frac {t}{2}}}{471691}+\frac {5 \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{11}+\cos \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \] Verified OK.

21.4.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}y^{\prime }+y^{\prime }+3 y=\left (1-\mathit {Heaviside}\left (t -2\right )\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}} \sin \left (t -2\right ), y \left (0\right )=1, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=2\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d t}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d t}y^{\prime }=-3 y-{\mathrm e}^{-\frac {t}{10}+\frac {1}{5}} \sin \left (t -2\right ) \mathit {Heaviside}\left (t -2\right )+{\mathrm e}^{-\frac {t}{10}+\frac {1}{5}} \sin \left (t -2\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} \\ {} & {} & \frac {d}{d t}y^{\prime }+y^{\prime }+3 y=-\left (-1+\mathit {Heaviside}\left (t -2\right )\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}} \sin \left (t -2\right ) \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+r +3=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {\left (-1\right )\pm \left (\sqrt {-11}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {11}}{2}, -\frac {1}{2}+\frac {\mathrm {I} \sqrt {11}}{2}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )=\cos \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )={\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, 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} \cos \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}+c_{2} {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, 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 -2\right )\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}} \sin \left (t -2\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} \cos \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} & {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \\ -\frac {\sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{2}-\frac {\cos \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{2} & -\frac {{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{2}+\frac {{\mathrm e}^{-\frac {t}{2}} \sqrt {11}\, \cos \left (\frac {\sqrt {11}\, 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 {11}\, {\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 {11}\, {\mathrm e}^{-\frac {t}{2}} \left (\sin \left (\frac {\sqrt {11}\, t}{2}\right ) \left (\int {\mathrm e}^{\frac {2 t}{5}+\frac {1}{5}} \sin \left (t -2\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right ) \left (-1+\mathit {Heaviside}\left (t -2\right )\right )d t \right )-\cos \left (\frac {\sqrt {11}\, t}{2}\right ) \left (\int {\mathrm e}^{\frac {2 t}{5}+\frac {1}{5}} \sin \left (t -2\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \left (-1+\mathit {Heaviside}\left (t -2\right )\right )d t \right )\right )}{11} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (t \right )=\frac {31800 \,{\mathrm e}^{-\frac {t}{2}} \left (-\frac {440 \mathit {Heaviside}\left (t -2\right ) {\mathrm e} \cos \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right )}{159}+\mathit {Heaviside}\left (t -2\right ) {\mathrm e} \sin \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right ) \sqrt {11}+\frac {440 \left (-1+\mathit {Heaviside}\left (t -2\right )\right ) \left (\cos \left (t -2\right )-\frac {191 \sin \left (t -2\right )}{80}\right ) {\mathrm e}^{\frac {2 t}{5}+\frac {1}{5}}}{159}\right )}{471691} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} \cos \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}+c_{2} {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right )+\frac {31800 \,{\mathrm e}^{-\frac {t}{2}} \left (-\frac {440 \mathit {Heaviside}\left (t -2\right ) {\mathrm e} \cos \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right )}{159}+\mathit {Heaviside}\left (t -2\right ) {\mathrm e} \sin \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right ) \sqrt {11}+\frac {440 \left (-1+\mathit {Heaviside}\left (t -2\right )\right ) \left (\cos \left (t -2\right )-\frac {191 \sin \left (t -2\right )}{80}\right ) {\mathrm e}^{\frac {2 t}{5}+\frac {1}{5}}}{159}\right )}{471691} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} \cos \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}+c_{2} {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right )+\frac {31800 {\mathrm e}^{-\frac {t}{2}} \left (-\frac {440 \mathit {Heaviside}\left (t -2\right ) {\mathrm e} \cos \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right )}{159}+\mathit {Heaviside}\left (t -2\right ) {\mathrm e} \sin \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right ) \sqrt {11}+\frac {440 \left (-1+\mathit {Heaviside}\left (t -2\right )\right ) \left (\cos \left (t -2\right )-\frac {191 \sin \left (t -2\right )}{80}\right ) {\mathrm e}^{\frac {2 t}{5}+\frac {1}{5}}}{159}\right )}{471691} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=1 \\ {} & {} & 1=c_{1} -\frac {8000 \left (\cos \left (2\right )+\frac {191 \sin \left (2\right )}{80}\right ) {\mathrm e}^{\frac {1}{5}}}{42881} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {c_{1} \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{2}-\frac {c_{1} \cos \left (\frac {\sqrt {11}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{2}+\frac {c_{2} {\mathrm e}^{-\frac {t}{2}} \sqrt {11}\, \cos \left (\frac {\sqrt {11}\, t}{2}\right )}{2}-\frac {15900 \,{\mathrm e}^{-\frac {t}{2}} \left (-\frac {440 \mathit {Heaviside}\left (t -2\right ) {\mathrm e} \cos \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right )}{159}+\mathit {Heaviside}\left (t -2\right ) {\mathrm e} \sin \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right ) \sqrt {11}+\frac {440 \left (-1+\mathit {Heaviside}\left (t -2\right )\right ) \left (\cos \left (t -2\right )-\frac {191 \sin \left (t -2\right )}{80}\right ) {\mathrm e}^{\frac {2 t}{5}+\frac {1}{5}}}{159}\right )}{471691}+\frac {31800 \,{\mathrm e}^{-\frac {t}{2}} \left (-\frac {440 \mathit {Dirac}\left (t -2\right ) {\mathrm e} \cos \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right )}{159}+\frac {220 \mathit {Heaviside}\left (t -2\right ) {\mathrm e} \sin \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right ) \sqrt {11}}{159}+\mathit {Dirac}\left (t -2\right ) {\mathrm e} \sin \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right ) \sqrt {11}+\frac {11 \mathit {Heaviside}\left (t -2\right ) {\mathrm e} \cos \left (\frac {\sqrt {11}\, \left (t -2\right )}{2}\right )}{2}+\frac {440 \mathit {Dirac}\left (t -2\right ) \left (\cos \left (t -2\right )-\frac {191 \sin \left (t -2\right )}{80}\right ) {\mathrm e}^{\frac {2 t}{5}+\frac {1}{5}}}{159}+\frac {440 \left (-1+\mathit {Heaviside}\left (t -2\right )\right ) \left (-\sin \left (t -2\right )-\frac {191 \cos \left (t -2\right )}{80}\right ) {\mathrm e}^{\frac {2 t}{5}+\frac {1}{5}}}{159}+\frac {176 \left (-1+\mathit {Heaviside}\left (t -2\right )\right ) \left (\cos \left (t -2\right )-\frac {191 \sin \left (t -2\right )}{80}\right ) {\mathrm e}^{\frac {2 t}{5}+\frac {1}{5}}}{159}\right )}{471691} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=2 \\ {} & {} & 2=-\frac {c_{1}}{2}+\frac {c_{2} \sqrt {11}}{2}+\frac {800 \left (\cos \left (2\right )+\frac {191 \sin \left (2\right )}{80}\right ) {\mathrm e}^{\frac {1}{5}}}{42881}-\frac {8000 \left (\sin \left (2\right )-\frac {191 \cos \left (2\right )}{80}\right ) {\mathrm e}^{\frac {1}{5}}}{42881} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =\frac {8000 \,{\mathrm e}^{\frac {1}{5}} \cos \left (2\right )}{42881}+\frac {19100 \,{\mathrm e}^{\frac {1}{5}} \sin \left (2\right )}{42881}+1, c_{2} =-\frac {5 \left (6360 \,{\mathrm e}^{\frac {1}{5}} \cos \left (2\right )-6256 \,{\mathrm e}^{\frac {1}{5}} \sin \left (2\right )-42881\right ) \sqrt {11}}{471691}\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=-\frac {8000 \,{\mathrm e}^{-\frac {t}{2}} \left (-\left (\left (\cos \left (t \right )-\frac {191 \sin \left (t \right )}{80}\right ) \cos \left (2\right )+\frac {191 \sin \left (2\right ) \left (\cos \left (t \right )+\frac {80 \sin \left (t \right )}{191}\right )}{80}\right ) \left (-1+\mathit {Heaviside}\left (t -2\right )\right ) {\mathrm e}^{\frac {2 t}{5}+\frac {1}{5}}+\left ({\mathrm e} \left (\frac {159 \sin \left (\sqrt {11}\right ) \sqrt {11}}{440}+\cos \left (\sqrt {11}\right )\right ) \mathit {Heaviside}\left (t -2\right )-\frac {42881}{8000}+\left (-\cos \left (2\right )-\frac {191 \sin \left (2\right )}{80}\right ) {\mathrm e}^{\frac {1}{5}}\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )-\frac {159 \left ({\mathrm e} \left (\sqrt {11}\, \cos \left (\sqrt {11}\right )-\frac {440 \sin \left (\sqrt {11}\right )}{159}\right ) \mathit {Heaviside}\left (t -2\right )-\left (-\frac {42881}{6360}+\left (\cos \left (2\right )-\frac {782 \sin \left (2\right )}{795}\right ) {\mathrm e}^{\frac {1}{5}}\right ) \sqrt {11}\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{440}\right )}{42881} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=-\frac {8000 \,{\mathrm e}^{-\frac {t}{2}} \left (-\left (\left (\cos \left (t \right )-\frac {191 \sin \left (t \right )}{80}\right ) \cos \left (2\right )+\frac {191 \sin \left (2\right ) \left (\cos \left (t \right )+\frac {80 \sin \left (t \right )}{191}\right )}{80}\right ) \left (-1+\mathit {Heaviside}\left (t -2\right )\right ) {\mathrm e}^{\frac {2 t}{5}+\frac {1}{5}}+\left ({\mathrm e} \left (\frac {159 \sin \left (\sqrt {11}\right ) \sqrt {11}}{440}+\cos \left (\sqrt {11}\right )\right ) \mathit {Heaviside}\left (t -2\right )-\frac {42881}{8000}+\left (-\cos \left (2\right )-\frac {191 \sin \left (2\right )}{80}\right ) {\mathrm e}^{\frac {1}{5}}\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )-\frac {159 \left ({\mathrm e} \left (\sqrt {11}\, \cos \left (\sqrt {11}\right )-\frac {440 \sin \left (\sqrt {11}\right )}{159}\right ) \mathit {Heaviside}\left (t -2\right )-\left (-\frac {42881}{6360}+\left (\cos \left (2\right )-\frac {782 \sin \left (2\right )}{795}\right ) {\mathrm e}^{\frac {1}{5}}\right ) \sqrt {11}\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{440}\right )}{42881} \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: 6.812 (sec). Leaf size: 178

dsolve([diff(y(t),t$2)+diff(y(t),t)+3*y(t)=(1-Heaviside(t-2))*exp(-(t-2)/10)*sin(t-2),y(0) = 1, D(y)(0) = 2],y(t), singsol=all)
 

\[ y \left (t \right ) = \frac {8000 \left (\left (\cos \left (t \right )-\frac {191 \sin \left (t \right )}{80}\right ) \cos \left (2\right )+\frac {191 \sin \left (2\right ) \left (\cos \left (t \right )+\frac {80 \sin \left (t \right )}{191}\right )}{80}\right ) \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-\frac {t}{10}+\frac {1}{5}}}{42881}+\frac {100 \left (11 \left (80 \cos \left (2\right )+191 \sin \left (2\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )-318 \left (\cos \left (2\right )-\frac {782 \sin \left (2\right )}{795}\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \sqrt {11}\right ) {\mathrm e}^{\frac {1}{5}-\frac {t}{2}}}{471691}+\left (-\frac {4000}{42881}+\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}-i\right ) \left (t -2\right )}+\left (-\frac {4000}{42881}-\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}+i\right ) \left (t -2\right )}+\frac {200 \operatorname {Heaviside}\left (t -2\right ) \left (\left (-159 \sqrt {11}\, \sin \left (\sqrt {11}\right )-440 \cos \left (\sqrt {11}\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (159 \cos \left (\sqrt {11}\right ) \sqrt {11}-440 \sin \left (\sqrt {11}\right )\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right )\right ) {\mathrm e}^{1-\frac {t}{2}}}{471691}+\frac {5 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{11}+{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right ) \]

✓ Solution by Mathematica

Time used: 6.103 (sec). Leaf size: 243

DSolve[{y''[t]+y'[t]+8*y[t]==(1-UnitStep[t-2])*Exp[-(t-2)/10]*Sin[t-2],{y[0]==1,y'[0]==2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {e^{-t/2} \left (-248000 e^{\frac {2 t}{5}+\frac {1}{5}} \cos (2-t)+5 \left (\sqrt {31} \left (483881-8 \sqrt [5]{e} (3295 \cos (2)-1782 \sin (2))\right ) \sin \left (\frac {\sqrt {31} t}{2}\right )-428420 e^{\frac {2 t}{5}+\frac {1}{5}} \sin (2-t)\right )+31 \cos \left (\frac {\sqrt {31} t}{2}\right ) \left (483881+100 \sqrt [5]{e} (80 \cos (2)+691 \sin (2))\right )\right )}{15000311} & t\leq 2 \\ \frac {e^{-t/2} \left (-248000 e \cos \left (\frac {1}{2} \sqrt {31} (t-2)\right )+5 \sqrt {31} \left (26360 e \sin \left (\frac {1}{2} \sqrt {31} (t-2)\right )+\left (483881-8 \sqrt [5]{e} (3295 \cos (2)-1782 \sin (2))\right ) \sin \left (\frac {\sqrt {31} t}{2}\right )\right )+31 \cos \left (\frac {\sqrt {31} t}{2}\right ) \left (483881+100 \sqrt [5]{e} (80 \cos (2)+691 \sin (2))\right )\right )}{15000311} & \text {True} \\ \end {array} \\ \end {array} \]