5.10 problem 10(c)

Internal problem ID [865]
Internal file name [OUTPUT/865_Sunday_June_05_2022_01_52_42_AM_21475340/index.tex]

Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section: Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number: 10(c).
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]]

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

5.10.1 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) &={\frac {1}{4}}\\ q(t) &=1\\ F &=\delta \left (-1+t \right ) \end {align*}

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

The domain of \(p(t)={\frac {1}{4}}\) 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 )+\frac {s Y \left (s \right )}{4}-\frac {y \left (0\right )}{4}+Y \left (s \right ) = {\mathrm e}^{-s}\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 )+\frac {s Y \left (s \right )}{4}+Y \left (s \right ) = {\mathrm e}^{-s} \end {align*}

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

Taking the inverse Laplace transform gives \begin {align*} y&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (\frac {4 \,{\mathrm e}^{-s}}{4 s^{2}+s +4}\right )\\ &= \frac {8 \operatorname {Heaviside}\left (-1+t \right ) \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, \left (-1+t \right )}{8}\right )}{21} \end {align*}

Converting the above solution to piecewise it becomes \[ y = \left \{\begin {array}{cc} 0 & t <1 \\ \frac {8 \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, \left (-1+t \right )}{8}\right )}{21} & 1\le t \end {array}\right . \] Simplifying the solution gives \[ y = \left \{\begin {array}{cc} 0 & t <1 \\ \frac {8 \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, \left (-1+t \right )}{8}\right )}{21} & 1\le t \end {array}\right . \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \left \{\begin {array}{cc} 0 & t <1 \\ \frac {8 \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, \left (-1+t \right )}{8}\right )}{21} & 1\le t \end {array}\right . \\ \end{align*}

Verification of solutions

\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ \frac {8 \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, \left (-1+t \right )}{8}\right )}{21} & 1\le t \end {array}\right . \] Verified OK.

5.10.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime \prime }+\frac {y^{\prime }}{4}+y=\mathit {Dirac}\left (-1+t \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 {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+\frac {1}{4} r +1=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {\left (-\frac {1}{4}\right )\pm \left (\sqrt {-\frac {63}{16}}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-\frac {1}{8}-\frac {3 \,\mathrm {I} \sqrt {7}}{8}, -\frac {1}{8}+\frac {3 \,\mathrm {I} \sqrt {7}}{8}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )={\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )={\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\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}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )+c_{2} {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\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 (-1+t \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}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) & {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \\ -\frac {{\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}-\frac {3 \,{\mathrm e}^{-\frac {t}{8}} \sqrt {7}\, \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8} & -\frac {{\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}+\frac {3 \,{\mathrm e}^{-\frac {t}{8}} \sqrt {7}\, \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )=\frac {3 \sqrt {7}\, {\mathrm e}^{-\frac {t}{4}}}{8} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (t \right ) \\ {} & {} & y_{p}\left (t \right )=\frac {8 \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \left (\int \mathit {Dirac}\left (-1+t \right )d t \right ) \left (\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \cos \left (\frac {3 \sqrt {7}}{8}\right )-\cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \sin \left (\frac {3 \sqrt {7}}{8}\right )\right )}{21} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (t \right )=\frac {8 \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \mathit {Heaviside}\left (-1+t \right ) \left (\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \cos \left (\frac {3 \sqrt {7}}{8}\right )-\cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \sin \left (\frac {3 \sqrt {7}}{8}\right )\right )}{21} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )+c_{2} {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )+\frac {8 \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \mathit {Heaviside}\left (-1+t \right ) \left (\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \cos \left (\frac {3 \sqrt {7}}{8}\right )-\cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \sin \left (\frac {3 \sqrt {7}}{8}\right )\right )}{21} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} {\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )+c_{2} {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )+\frac {8 \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \mathit {Heaviside}\left (-1+t \right ) \left (\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \cos \left (\frac {3 \sqrt {7}}{8}\right )-\cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \sin \left (\frac {3 \sqrt {7}}{8}\right )\right )}{21} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=c_{1} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {c_{1} {\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}-\frac {3 c_{1} {\mathrm e}^{-\frac {t}{8}} \sqrt {7}\, \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}-\frac {c_{2} {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}+\frac {3 c_{2} {\mathrm e}^{-\frac {t}{8}} \sqrt {7}\, \cos \left (\frac {3 \sqrt {7}\, t}{8}\right )}{8}-\frac {\sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \mathit {Heaviside}\left (-1+t \right ) \left (\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \cos \left (\frac {3 \sqrt {7}}{8}\right )-\cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \sin \left (\frac {3 \sqrt {7}}{8}\right )\right )}{21}+\frac {8 \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \mathit {Dirac}\left (-1+t \right ) \left (\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \cos \left (\frac {3 \sqrt {7}}{8}\right )-\cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \sin \left (\frac {3 \sqrt {7}}{8}\right )\right )}{21}+\frac {8 \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \mathit {Heaviside}\left (-1+t \right ) \left (\frac {3 \sqrt {7}\, \cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \cos \left (\frac {3 \sqrt {7}}{8}\right )}{8}+\frac {3 \sqrt {7}\, \sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \sin \left (\frac {3 \sqrt {7}}{8}\right )}{8}\right )}{21} \\ {} & \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}}{8}+\frac {3 \sqrt {7}\, c_{2}}{8} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =0, c_{2} =0\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {8 \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \mathit {Heaviside}\left (-1+t \right ) \left (\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \cos \left (\frac {3 \sqrt {7}}{8}\right )-\cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \sin \left (\frac {3 \sqrt {7}}{8}\right )\right )}{21} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {8 \sqrt {7}\, {\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \mathit {Heaviside}\left (-1+t \right ) \left (\sin \left (\frac {3 \sqrt {7}\, t}{8}\right ) \cos \left (\frac {3 \sqrt {7}}{8}\right )-\cos \left (\frac {3 \sqrt {7}\, t}{8}\right ) \sin \left (\frac {3 \sqrt {7}}{8}\right )\right )}{21} \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: 1.907 (sec). Leaf size: 28

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

\[ y \left (t \right ) = \frac {8 \,{\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \operatorname {Heaviside}\left (t -1\right ) \sqrt {7}\, \sin \left (\frac {3 \sqrt {7}\, \left (t -1\right )}{8}\right )}{21} \]

✓ Solution by Mathematica

Time used: 0.075 (sec). Leaf size: 42

DSolve[{y''[t]+1/4*y'[t]+y[t]==DiracDelta[t-1],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {8 e^{\frac {1}{8}-\frac {t}{8}} \theta (t-1) \sin \left (\frac {3}{8} \sqrt {7} (t-1)\right )}{3 \sqrt {7}} \]