4.19 problem Problem 3(e)

Internal problem ID [12326]
Internal file name [OUTPUT/10979_Monday_October_02_2023_02_47_41_AM_384554/index.tex]

Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section: Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number: Problem 3(e).
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]]

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

4.19.1 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) &=2\\ q(t) &=2\\ F &=5 \cos \left (t \right ) \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }+2 y^{\prime }+2 y = 5 \cos \left (t \right ) \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \end {align*}

The domain of \(p(t)=2\) 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 )+2 s Y \left (s \right )-2 y \left (0\right )+2 Y \left (s \right ) = \frac {5 \,{\mathrm e}^{-\frac {s \pi }{2}}+5 s}{s^{2}+1}\tag {1} \end {align*}

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

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

Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {s^{3}+s^{2}+5 \,{\mathrm e}^{-\frac {s \pi }{2}}+6 s +1}{\left (s^{2}+1\right ) \left (s^{2}+2 s +2\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^{3}+s^{2}+5 \,{\mathrm e}^{-\frac {s \pi }{2}}+6 s +1}{\left (s^{2}+1\right ) \left (s^{2}+2 s +2\right )}\right )\\ &= \frac {7 \cos \left (t \right )}{5}+\frac {9 \sin \left (t \right )}{5}-2 \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}} \left (\cos \left (t \right ) \cosh \left (\frac {t}{2}-\frac {\pi }{4}\right )+2 \sin \left (t \right ) \sinh \left (\frac {t}{2}-\frac {\pi }{4}\right )\right )-\frac {2 \,{\mathrm e}^{-t} \left (\cos \left (t \right )+8 \sin \left (t \right )\right )}{5}+\frac {2 \left (-2 \cos \left (t \right ) \sinh \left (\frac {t}{2}\right )+\sin \left (t \right ) \cosh \left (\frac {t}{2}\right )\right ) {\mathrm e}^{-\frac {t}{2}}}{5} \end {align*}

Converting the above solution to piecewise it becomes \[ y = \left \{\begin {array}{cc} \frac {7 \cos \left (t \right )}{5}+\frac {9 \sin \left (t \right )}{5}-\frac {2 \,{\mathrm e}^{-t} \left (\cos \left (t \right )+8 \sin \left (t \right )\right )}{5}+\frac {2 \left (-2 \cos \left (t \right ) \sinh \left (\frac {t}{2}\right )+\sin \left (t \right ) \cosh \left (\frac {t}{2}\right )\right ) {\mathrm e}^{-\frac {t}{2}}}{5} & t <\frac {\pi }{2} \\ \frac {7 \cos \left (t \right )}{5}+\frac {9 \sin \left (t \right )}{5}-\frac {2 \,{\mathrm e}^{-t} \left (\cos \left (t \right )+8 \sin \left (t \right )\right )}{5}+\frac {2 \left (-2 \cos \left (t \right ) \sinh \left (\frac {t}{2}\right )+\sin \left (t \right ) \cosh \left (\frac {t}{2}\right )\right ) {\mathrm e}^{-\frac {t}{2}}}{5}-2 \,{\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}} \left (\cos \left (t \right ) \cosh \left (\frac {t}{2}-\frac {\pi }{4}\right )+2 \sin \left (t \right ) \sinh \left (\frac {t}{2}-\frac {\pi }{4}\right )\right ) & \frac {\pi }{2}\le t \end {array}\right . \] Simplifying the solution gives \[ y = -3 \,{\mathrm e}^{-t} \sin \left (t \right )-\left (\left \{\begin {array}{cc} -\cos \left (t \right )-2 \sin \left (t \right ) & t <\frac {\pi }{2} \\ {\mathrm e}^{-t +\frac {\pi }{2}} \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) & \frac {\pi }{2}\le t \end {array}\right .\right ) \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= -3 \,{\mathrm e}^{-t} \sin \left (t \right )-\left (\left \{\begin {array}{cc} -\cos \left (t \right )-2 \sin \left (t \right ) & t <\frac {\pi }{2} \\ {\mathrm e}^{-t +\frac {\pi }{2}} \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) & \frac {\pi }{2}\le t \end {array}\right .\right ) \\ \end{align*}

Verification of solutions

\[ y = -3 \,{\mathrm e}^{-t} \sin \left (t \right )-\left (\left \{\begin {array}{cc} -\cos \left (t \right )-2 \sin \left (t \right ) & t <\frac {\pi }{2} \\ {\mathrm e}^{-t +\frac {\pi }{2}} \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) & \frac {\pi }{2}\le t \end {array}\right .\right ) \] Verified OK.

4.19.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}y^{\prime }+2 y^{\prime }+2 y=\left (5 \mathit {Heaviside}\left (t \right )-5 \mathit {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \cos \left (t \right ), y \left (0\right )=1, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=-1\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 }=-2 y+5 \cos \left (t \right ) \mathit {Heaviside}\left (t \right )-5 \cos \left (t \right ) \mathit {Heaviside}\left (t -\frac {\pi }{2}\right )-2 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 }+2 y^{\prime }+2 y=5 \cos \left (t \right ) \left (\mathit {Heaviside}\left (t \right )-\mathit {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+2 r +2=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {\left (-2\right )\pm \left (\sqrt {-4}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-1-\mathrm {I}, -1+\mathrm {I}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )={\mathrm e}^{-t} \cos \left (t \right ) \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )={\mathrm e}^{-t} \sin \left (t \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}^{-t} \cos \left (t \right )+c_{2} {\mathrm e}^{-t} \sin \left (t \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 )=5 \cos \left (t \right ) \left (\mathit {Heaviside}\left (t \right )-\mathit {Heaviside}\left (t -\frac {\pi }{2}\right )\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}^{-t} \cos \left (t \right ) & {\mathrm e}^{-t} \sin \left (t \right ) \\ -{\mathrm e}^{-t} \cos \left (t \right )-{\mathrm e}^{-t} \sin \left (t \right ) & -{\mathrm e}^{-t} \sin \left (t \right )+{\mathrm e}^{-t} \cos \left (t \right ) \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )={\mathrm e}^{-2 t} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (t \right ) \\ {} & {} & y_{p}\left (t \right )=-\frac {5 \,{\mathrm e}^{-t} \cos \left (t \right ) \left (\int \left (\mathit {Heaviside}\left (t \right )-\mathit {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \sin \left (2 t \right ) {\mathrm e}^{t}d t \right )}{2}+5 \,{\mathrm e}^{-t} \sin \left (t \right ) \left (\int \cos \left (t \right )^{2} {\mathrm e}^{t} \left (\mathit {Heaviside}\left (t \right )-\mathit {Heaviside}\left (t -\frac {\pi }{2}\right )\right )d t \right ) \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (t \right )=-\mathit {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) {\mathrm e}^{-t +\frac {\pi }{2}}+\mathit {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (-\cos \left (t \right )-2 \sin \left (t \right )\right )-\mathit {Heaviside}\left (t \right ) \left (\left (\cos \left (t \right )+3 \sin \left (t \right )\right ) {\mathrm e}^{-t}-\cos \left (t \right )-2 \sin \left (t \right )\right ) \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{-t} \cos \left (t \right )+c_{2} {\mathrm e}^{-t} \sin \left (t \right )-\mathit {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) {\mathrm e}^{-t +\frac {\pi }{2}}+\mathit {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (-\cos \left (t \right )-2 \sin \left (t \right )\right )-\mathit {Heaviside}\left (t \right ) \left (\left (\cos \left (t \right )+3 \sin \left (t \right )\right ) {\mathrm e}^{-t}-\cos \left (t \right )-2 \sin \left (t \right )\right ) \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} {\mathrm e}^{-t} \cos \left (t \right )+c_{2} {\mathrm e}^{-t} \sin \left (t \right )-\mathit {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) {\mathrm e}^{-t +\frac {\pi }{2}}+\mathit {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (-\cos \left (t \right )-2 \sin \left (t \right )\right )-\mathit {Heaviside}\left (t \right ) \left (\left (\cos \left (t \right )+3 \sin \left (t \right )\right ) {\mathrm e}^{-t}-\cos \left (t \right )-2 \sin \left (t \right )\right ) \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=1 \\ {} & {} & 1=c_{1} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=-c_{1} {\mathrm e}^{-t} \cos \left (t \right )-c_{1} {\mathrm e}^{-t} \sin \left (t \right )-c_{2} {\mathrm e}^{-t} \sin \left (t \right )+c_{2} {\mathrm e}^{-t} \cos \left (t \right )-\mathit {Dirac}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) {\mathrm e}^{-t +\frac {\pi }{2}}-\mathit {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (-\sin \left (t \right )-2 \cos \left (t \right )\right ) {\mathrm e}^{-t +\frac {\pi }{2}}+\mathit {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) {\mathrm e}^{-t +\frac {\pi }{2}}+\mathit {Dirac}\left (t -\frac {\pi }{2}\right ) \left (-\cos \left (t \right )-2 \sin \left (t \right )\right )+\mathit {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\sin \left (t \right )-2 \cos \left (t \right )\right )-\mathit {Dirac}\left (t \right ) \left (\left (\cos \left (t \right )+3 \sin \left (t \right )\right ) {\mathrm e}^{-t}-\cos \left (t \right )-2 \sin \left (t \right )\right )-\mathit {Heaviside}\left (t \right ) \left (\left (3 \cos \left (t \right )-\sin \left (t \right )\right ) {\mathrm e}^{-t}-\left (\cos \left (t \right )+3 \sin \left (t \right )\right ) {\mathrm e}^{-t}+\sin \left (t \right )-2 \cos \left (t \right )\right ) \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=-1 \\ {} & {} & -1=-c_{1} +c_{2} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =1, c_{2} =0\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=-\mathit {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) {\mathrm e}^{-t +\frac {\pi }{2}}+\mathit {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (-\cos \left (t \right )-2 \sin \left (t \right )\right )+\left (\left (1-\mathit {Heaviside}\left (t \right )\right ) \cos \left (t \right )-3 \sin \left (t \right ) \mathit {Heaviside}\left (t \right )\right ) {\mathrm e}^{-t}+\mathit {Heaviside}\left (t \right ) \left (\cos \left (t \right )+2 \sin \left (t \right )\right ) \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=-\mathit {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) {\mathrm e}^{-t +\frac {\pi }{2}}+\mathit {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (-\cos \left (t \right )-2 \sin \left (t \right )\right )+\left (\left (1-\mathit {Heaviside}\left (t \right )\right ) \cos \left (t \right )-3 \sin \left (t \right ) \mathit {Heaviside}\left (t \right )\right ) {\mathrm e}^{-t}+\mathit {Heaviside}\left (t \right ) \left (\cos \left (t \right )+2 \sin \left (t \right )\right ) \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: 5.734 (sec). Leaf size: 88

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+2*y(t)=5*cos(t)*(Heaviside(t)-Heaviside(t-Pi/2)),y(0) = 1, D(y)(0) = -1],y(t), singsol=all)
 

\[ y \left (t \right ) = -\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) {\mathrm e}^{-t +\frac {\pi }{2}}+\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (-\cos \left (t \right )-2 \sin \left (t \right )\right )-3 \sin \left (t \right ) {\mathrm e}^{-t}+\cos \left (t \right )+2 \sin \left (t \right ) \]

✓ Solution by Mathematica

Time used: 0.095 (sec). Leaf size: 72

DSolve[{y''[t]+2*y'[t]+2*y[t]==5*Cos[t]*(UnitStep[t]-UnitStep[t-Pi/2]),{y[0]==1,y'[0]==-1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-t} \cos (t) & t<0 \\ e^{-t} \left (\left (-3+2 e^{\pi /2}\right ) \sin (t)-e^{\pi /2} \cos (t)\right ) & 2 t>\pi \\ \cos (t)+\left (2-3 e^{-t}\right ) \sin (t) & \text {True} \\ \end {array} \\ \end {array} \]