problem 6

Internal problem ID [6702]
Internal ﬁle name [OUTPUT/5950_Sunday_June_05_2022_04_03_40_PM_50583576/index.tex]

Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section: CHAPTER 7 THE LAPLACE TRANSFORM. EXERCISES 7.5. Page 315
Problem number: 6.
ODE order: 2.
ODE degree: 1.

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

\[ \boxed {y^{\prime \prime }+y=\delta \left (t -2 \pi \right )+\delta \left (t -4 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}

8.6.1 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) &=0\\ q(t) &=1\\ F &=\delta \left (t -2 \pi \right )+\delta \left (t -4 \pi \right ) \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }+y = \delta \left (t -2 \pi \right )+\delta \left (t -4 \pi \right ) \end {align*}

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 )+Y \left (s \right ) = {\mathrm e}^{-2 s \pi }+{\mathrm e}^{-4 s \pi }\tag {1} \end {align*}

But the initial conditions are \begin {align*} y \left (0\right )&=1\\ 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 ) = {\mathrm e}^{-2 s \pi }+{\mathrm e}^{-4 s \pi } \end {align*}

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

Taking the inverse Laplace transform gives \begin {align*} y&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (\frac {{\mathrm e}^{-2 s \pi }+{\mathrm e}^{-4 s \pi }+s}{s^{2}+1}\right )\\ &= \cos \left (t \right )+\sin \left (t \right ) \left (\operatorname {Heaviside}\left (t -2 \pi \right )+\operatorname {Heaviside}\left (t -4 \pi \right )\right ) \end {align*}

Converting the above solution to piecewise it becomes \[ y = \left \{\begin {array}{cc} \cos \left (t \right ) & t <2 \pi \\ \cos \left (t \right )+\sin \left (t \right ) & t <4 \pi \\ \cos \left (t \right )+2 \sin \left (t \right ) & 4 \pi \le t \end {array}\right . \] Simplifying the solution gives \[ y = \cos \left (t \right )+\sin \left (t \right ) \left (\left \{\begin {array}{cc} 0 & t <2 \pi \\ 1 & t <4 \pi \\ 2 & 4 \pi \le t \end {array}\right .\right ) \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \cos \left (t \right )+\sin \left (t \right ) \left (\left \{\begin {array}{cc} 0 & t <2 \pi \\ 1 & t <4 \pi \\ 2 & 4 \pi \le t \end {array}\right .\right ) \\ \end{align*}

Veriﬁcation of solutions

\[ y = \cos \left (t \right )+\sin \left (t \right ) \left (\left \{\begin {array}{cc} 0 & t <2 \pi \\ 1 & t <4 \pi \\ 2 & 4 \pi \le t \end {array}\right .\right ) \] Veriﬁed OK.

8.6.2 Maple step by step solution

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

\[ y \left (t \right ) = \sin \left (t \right ) \operatorname {Heaviside}\left (t -2 \pi \right )+\sin \left (t \right ) \operatorname {Heaviside}\left (t -4 \pi \right )+\cos \left (t \right ) \]

8.6 problem 6

8.6.1 Existence and uniqueness analysis

8.6.2 Maple step by step solution