21.2 problem 2

Internal problem ID [13235]
Internal file name [OUTPUT/11891_Tuesday_December_05_2023_12_12_48_PM_64669449/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: 2.
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]]

\[ \boxed {y^{\prime \prime }+y^{\prime }+5 y=\operatorname {Heaviside}\left (t -2\right ) \sin \left (4 t -8\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 0] \end {align*}

21.2.1 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) &=5\\ F &=\operatorname {Heaviside}\left (t -2\right ) \sin \left (4 t -8\right ) \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }+y^{\prime }+5 y = \operatorname {Heaviside}\left (t -2\right ) \sin \left (4 t -8\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 )+5 Y \left (s \right ) = \frac {4 \,{\mathrm e}^{-2 s}}{s^{2}+16}\tag {1} \end {align*}

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

Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )+2+2 s +s Y \left (s \right )+5 Y \left (s \right ) = \frac {4 \,{\mathrm e}^{-2 s}}{s^{2}+16} \end {align*}

Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {-2 s^{3}-2 s^{2}+4 \,{\mathrm e}^{-2 s}-32 s -32}{\left (s^{2}+16\right ) \left (s^{2}+s +5\right )} \end {align*}

Taking the inverse Laplace transform gives \begin {align*} y&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (\frac {-2 s^{3}-2 s^{2}+4 \,{\mathrm e}^{-2 s}-32 s -32}{\left (s^{2}+16\right ) \left (s^{2}+s +5\right )}\right )\\ &= -\frac {2 \,{\mathrm e}^{-\frac {t}{2}} \left (\sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right )+19 \cos \left (\frac {\sqrt {19}\, t}{2}\right )\right )}{19}+\frac {\left (-209 \sin \left (4 t -8\right )-76 \cos \left (4 t -8\right )+4 \,{\mathrm e}^{-\frac {t}{2}+1} \left (23 \sqrt {19}\, \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right )+19 \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right )\right )\right ) \operatorname {Heaviside}\left (t -2\right )}{2603} \end {align*}

Converting the above solution to piecewise it becomes \[ y = \left \{\begin {array}{cc} -\frac {2 \,{\mathrm e}^{-\frac {t}{2}} \left (\sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right )+19 \cos \left (\frac {\sqrt {19}\, t}{2}\right )\right )}{19} & t <2 \\ -\frac {2 \,{\mathrm e}^{-\frac {t}{2}} \left (\sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right )+19 \cos \left (\frac {\sqrt {19}\, t}{2}\right )\right )}{19}-\frac {11 \sin \left (4 t -8\right )}{137}-\frac {4 \cos \left (4 t -8\right )}{137}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}+1} \left (23 \sqrt {19}\, \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right )+19 \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right )\right )}{2603} & 2\le t \end {array}\right . \] Simplifying the solution gives \[ y = \left \{\begin {array}{cc} -\frac {2 \,{\mathrm e}^{-\frac {t}{2}} \left (\sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right )+19 \cos \left (\frac {\sqrt {19}\, t}{2}\right )\right )}{19} & t <2 \\ -\frac {2 \sin \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {19}}{19}-2 \cos \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}-\frac {11 \sin \left (4 t -8\right )}{137}-\frac {4 \cos \left (4 t -8\right )}{137}+\frac {92 \,{\mathrm e}^{-\frac {t}{2}+1} \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \sqrt {19}}{2603}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}+1} \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right )}{137} & 2\le t \end {array}\right . \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \left \{\begin {array}{cc} -\frac {2 \,{\mathrm e}^{-\frac {t}{2}} \left (\sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right )+19 \cos \left (\frac {\sqrt {19}\, t}{2}\right )\right )}{19} & t <2 \\ -\frac {2 \sin \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {19}}{19}-2 \cos \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}-\frac {11 \sin \left (4 t -8\right )}{137}-\frac {4 \cos \left (4 t -8\right )}{137}+\frac {92 \,{\mathrm e}^{-\frac {t}{2}+1} \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \sqrt {19}}{2603}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}+1} \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right )}{137} & 2\le t \end {array}\right . \\ \end{align*}

Verification of solutions

\[ y = \left \{\begin {array}{cc} -\frac {2 \,{\mathrm e}^{-\frac {t}{2}} \left (\sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right )+19 \cos \left (\frac {\sqrt {19}\, t}{2}\right )\right )}{19} & t <2 \\ -\frac {2 \sin \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {19}}{19}-2 \cos \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}-\frac {11 \sin \left (4 t -8\right )}{137}-\frac {4 \cos \left (4 t -8\right )}{137}+\frac {92 \,{\mathrm e}^{-\frac {t}{2}+1} \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \sqrt {19}}{2603}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}+1} \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right )}{137} & 2\le t \end {array}\right . \] Verified OK.

21.2.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}y^{\prime }+y^{\prime }+5 y=\mathit {Heaviside}\left (t -2\right ) \sin \left (4 t -8\right ), y \left (0\right )=-2, 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 \\ {} & {} & \frac {d}{d t}y^{\prime } \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+r +5=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {\left (-1\right )\pm \left (\sqrt {-19}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {19}}{2}, -\frac {1}{2}+\frac {\mathrm {I} \sqrt {19}}{2}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )=\cos \left (\frac {\sqrt {19}\, 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 {19}\, 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 {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}+c_{2} {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {19}\, 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 )=\mathit {Heaviside}\left (t -2\right ) \sin \left (4 t -8\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 {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} & {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {19}\, t}{2}\right ) \\ -\frac {\sin \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {19}}{2}-\frac {\cos \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{2} & -\frac {{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {19}\, t}{2}\right )}{2}+\frac {{\mathrm e}^{-\frac {t}{2}} \sqrt {19}\, \cos \left (\frac {\sqrt {19}\, 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 {19}\, {\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 {19}\, {\mathrm e}^{-\frac {t}{2}} \left (\cos \left (\frac {\sqrt {19}\, t}{2}\right ) \left (\int {\mathrm e}^{\frac {t}{2}} \sin \left (4 t -8\right ) \mathit {Heaviside}\left (t -2\right ) \sin \left (\frac {\sqrt {19}\, t}{2}\right )d t \right )-\sin \left (\frac {\sqrt {19}\, t}{2}\right ) \left (\int {\mathrm e}^{\frac {t}{2}} \sin \left (4 t -8\right ) \mathit {Heaviside}\left (t -2\right ) \cos \left (\frac {\sqrt {19}\, t}{2}\right )d t \right )\right )}{19} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (t \right )=\frac {\mathit {Heaviside}\left (t -2\right ) \left (92 \,{\mathrm e}^{-\frac {t}{2}+1} \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \sqrt {19}-209 \sin \left (4 t -8\right )-76 \cos \left (4 t -8\right )+76 \,{\mathrm e}^{-\frac {t}{2}+1} \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right )\right )}{2603} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} \cos \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}+c_{2} {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {19}\, t}{2}\right )+\frac {\mathit {Heaviside}\left (t -2\right ) \left (92 \,{\mathrm e}^{-\frac {t}{2}+1} \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \sqrt {19}-209 \sin \left (4 t -8\right )-76 \cos \left (4 t -8\right )+76 \,{\mathrm e}^{-\frac {t}{2}+1} \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right )\right )}{2603} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} \cos \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}+c_{2} {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {19}\, t}{2}\right )+\frac {\mathit {Heaviside}\left (t -2\right ) \left (92 {\mathrm e}^{-\frac {t}{2}+1} \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \sqrt {19}-209 \sin \left (4 t -8\right )-76 \cos \left (4 t -8\right )+76 {\mathrm e}^{-\frac {t}{2}+1} \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right )\right )}{2603} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=-2 \\ {} & {} & -2=c_{1} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {c_{1} \sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{2}-\frac {c_{1} \cos \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {19}\, t}{2}\right )}{2}+\frac {c_{2} {\mathrm e}^{-\frac {t}{2}} \sqrt {19}\, \cos \left (\frac {\sqrt {19}\, t}{2}\right )}{2}+\frac {\mathit {Dirac}\left (t -2\right ) \left (92 \,{\mathrm e}^{-\frac {t}{2}+1} \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \sqrt {19}-209 \sin \left (4 t -8\right )-76 \cos \left (4 t -8\right )+76 \,{\mathrm e}^{-\frac {t}{2}+1} \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right )\right )}{2603}+\frac {\mathit {Heaviside}\left (t -2\right ) \left (-84 \,{\mathrm e}^{-\frac {t}{2}+1} \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \sqrt {19}+836 \,{\mathrm e}^{-\frac {t}{2}+1} \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right )-836 \cos \left (4 t -8\right )+304 \sin \left (4 t -8\right )\right )}{2603} \\ {} & \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 {19}}{2} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =-2, c_{2} =-\frac {2 \sqrt {19}}{19}\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {4 \mathit {Heaviside}\left (t -2\right ) \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) {\mathrm e}^{-\frac {t}{2}+1}}{137}+\frac {92 \sqrt {19}\, \mathit {Heaviside}\left (t -2\right ) \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) {\mathrm e}^{-\frac {t}{2}+1}}{2603}-2 \cos \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}-\frac {2 \sin \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {19}}{19}-\frac {4 \left (\cos \left (4 t -8\right )+\frac {11 \sin \left (4 t -8\right )}{4}\right ) \mathit {Heaviside}\left (t -2\right )}{137} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {4 \mathit {Heaviside}\left (t -2\right ) \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) {\mathrm e}^{-\frac {t}{2}+1}}{137}+\frac {92 \sqrt {19}\, \mathit {Heaviside}\left (t -2\right ) \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) {\mathrm e}^{-\frac {t}{2}+1}}{2603}-2 \cos \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}}-\frac {2 \sin \left (\frac {\sqrt {19}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {19}}{19}-\frac {4 \left (\cos \left (4 t -8\right )+\frac {11 \sin \left (4 t -8\right )}{4}\right ) \mathit {Heaviside}\left (t -2\right )}{137} \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: 6.703 (sec). Leaf size: 89

dsolve([diff(y(t),t$2)+diff(y(t),t)+5*y(t)=Heaviside(t-2)*sin(4*(t-2)),y(0) = -2, D(y)(0) = 0],y(t), singsol=all)
 

\[ y \left (t \right ) = \frac {4 \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{1-\frac {t}{2}}}{137}+\frac {92 \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \operatorname {Heaviside}\left (t -2\right ) \sqrt {19}\, {\mathrm e}^{1-\frac {t}{2}}}{2603}-2 \,{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {19}\, t}{2}\right )-\frac {2 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right )}{19}-\frac {4 \left (\cos \left (4 t -8\right )+\frac {11 \sin \left (4 t -8\right )}{4}\right ) \operatorname {Heaviside}\left (t -2\right )}{137} \]

✓ Solution by Mathematica

Time used: 6.103 (sec). Leaf size: 163

DSolve[{y''[t]+y'[t]+5*y[t]==UnitStep[t-2]*Sin[4*(t-2)],{y[0]==-2,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -\frac {2}{19} e^{-t/2} \left (19 \cos \left (\frac {\sqrt {19} t}{2}\right )+\sqrt {19} \sin \left (\frac {\sqrt {19} t}{2}\right )\right ) & t\leq 2 \\ \frac {e^{-t/2} \left (-76 e^{t/2} \cos (8-4 t)+76 e \cos \left (\frac {1}{2} \sqrt {19} (t-2)\right )-5206 \cos \left (\frac {\sqrt {19} t}{2}\right )+209 e^{t/2} \sin (8-4 t)+92 \sqrt {19} e \sin \left (\frac {1}{2} \sqrt {19} (t-2)\right )-274 \sqrt {19} \sin \left (\frac {\sqrt {19} t}{2}\right )\right )}{2603} & \text {True} \\ \end {array} \\ \end {array} \]