Internal problem ID [12337]
Internal file name [OUTPUT/10990_Monday_October_02_2023_02_47_44_AM_46001342/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 5(a).
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"
Maple gives the following as the ode type
[[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 \pi ^{2} y=3 \delta \left (t -\frac {1}{3}\right )-\delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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) &=4 \pi ^{2}\\ F &=3 \delta \left (t -\frac {1}{3}\right )-\delta \left (t -1\right ) \end {align*}
Hence the ode is \begin {align*} y^{\prime \prime }+4 \pi ^{2} y = 3 \delta \left (t -\frac {1}{3}\right )-\delta \left (t -1\right ) \end {align*}
The domain of \(p(t)=0\) 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 )+4 \pi ^{2} Y \left (s \right ) = 3 \,{\mathrm e}^{-\frac {s}{3}}-{\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 )+4 \pi ^{2} Y \left (s \right ) = 3 \,{\mathrm e}^{-\frac {s}{3}}-{\mathrm e}^{-s} \end {align*}
Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {3 \,{\mathrm e}^{-\frac {s}{3}}-{\mathrm e}^{-s}}{4 \pi ^{2}+s^{2}} \end {align*}
Taking the inverse Laplace transform gives \begin {align*} y&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (\frac {3 \,{\mathrm e}^{-\frac {s}{3}}-{\mathrm e}^{-s}}{4 \pi ^{2}+s^{2}}\right )\\ &= \frac {-\operatorname {Heaviside}\left (t -1\right ) \sin \left (2 \pi t \right )+3 \sin \left (\frac {2 \pi \left (3 t -1\right )}{3}\right ) \operatorname {Heaviside}\left (t -\frac {1}{3}\right )}{2 \pi } \end {align*}
Converting the above solution to piecewise it becomes \[
y = \left \{\begin {array}{cc} 0 & t <\frac {1}{3} \\ \frac {3 \sin \left (\frac {2 \pi \left (3 t -1\right )}{3}\right )}{2 \pi } & t <1 \\ \frac {-\sin \left (2 \pi t \right )+3 \sin \left (\frac {2 \pi \left (3 t -1\right )}{3}\right )}{2 \pi } & 1\le t \end {array}\right .
\] Simplifying the solution gives \[
y = -\frac {\left \{\begin {array}{cc} 0 & t <\frac {1}{3} \\ 3 \sin \left (\pi \left (2 t +\frac {1}{3}\right )\right ) & t <1 \\ \frac {3 \cos \left (2 \pi t \right ) \sqrt {3}}{2}+\frac {5 \sin \left (2 \pi t \right )}{2} & 1\le t \end {array}\right .}{2 \pi }
\]
The solution(s) found are the following \begin{align*}
\tag{1} y &= -\frac {\left \{\begin {array}{cc} 0 & t <\frac {1}{3} \\ 3 \sin \left (\pi \left (2 t +\frac {1}{3}\right )\right ) & t <1 \\ \frac {3 \cos \left (2 \pi t \right ) \sqrt {3}}{2}+\frac {5 \sin \left (2 \pi t \right )}{2} & 1\le t \end {array}\right .}{2 \pi } \\
\end{align*} Verification of solutions
\[
y = -\frac {\left \{\begin {array}{cc} 0 & t <\frac {1}{3} \\ 3 \sin \left (\pi \left (2 t +\frac {1}{3}\right )\right ) & t <1 \\ \frac {3 \cos \left (2 \pi t \right ) \sqrt {3}}{2}+\frac {5 \sin \left (2 \pi t \right )}{2} & 1\le t \end {array}\right .}{2 \pi }
\] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime \prime }+4 \pi ^{2} y=3 \mathit {Dirac}\left (t -\frac {1}{3}\right )-\mathit {Dirac}\left (t -1\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} \\ {} & {} & 4 \pi ^{2}+r^{2}=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {0\pm \left (\sqrt {-16 \pi ^{2}}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-2 \,\mathrm {I} \pi , 2 \,\mathrm {I} \pi \right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )=\cos \left (2 \pi t \right ) \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )=\sin \left (2 \pi 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 (2 \pi t \right )+c_{2} \sin \left (2 \pi 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 )=3 \mathit {Dirac}\left (t -\frac {1}{3}\right )-\mathit {Dirac}\left (t -1\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 (2 \pi t \right ) & \sin \left (2 \pi t \right ) \\ -2 \pi \sin \left (2 \pi t \right ) & 2 \pi \cos \left (2 \pi t \right ) \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )=2 \pi \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (t \right ) \\ {} & {} & y_{p}\left (t \right )=\frac {-3 \cos \left (2 \pi t \right ) \sqrt {3}\, \left (\int \mathit {Dirac}\left (t -\frac {1}{3}\right )d t \right )-\sin \left (2 \pi t \right ) \left (\int \left (2 \mathit {Dirac}\left (t -1\right )+3 \mathit {Dirac}\left (t -\frac {1}{3}\right )\right )d t \right )}{4 \pi } \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (t \right )=\frac {\left (-3 \cos \left (2 \pi t \right ) \sqrt {3}-3 \sin \left (2 \pi t \right )\right ) \mathit {Heaviside}\left (t -\frac {1}{3}\right )-2 \mathit {Heaviside}\left (t -1\right ) \sin \left (2 \pi t \right )}{4 \pi } \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} \cos \left (2 \pi t \right )+c_{2} \sin \left (2 \pi t \right )+\frac {\left (-3 \cos \left (2 \pi t \right ) \sqrt {3}-3 \sin \left (2 \pi t \right )\right ) \mathit {Heaviside}\left (t -\frac {1}{3}\right )-2 \mathit {Heaviside}\left (t -1\right ) \sin \left (2 \pi t \right )}{4 \pi } \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} \cos \left (2 \pi t \right )+c_{2} \sin \left (2 \pi t \right )+\frac {\left (-3 \cos \left (2 \pi t \right ) \sqrt {3}-3 \sin \left (2 \pi t \right )\right ) \mathit {Heaviside}\left (t -\frac {1}{3}\right )-2 \mathit {Heaviside}\left (t -1\right ) \sin \left (2 \pi t \right )}{4 \pi } \\ {} & \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 }=-2 c_{1} \pi \sin \left (2 \pi t \right )+2 c_{2} \pi \cos \left (2 \pi t \right )+\frac {\left (6 \pi \sin \left (2 \pi t \right ) \sqrt {3}-6 \pi \cos \left (2 \pi t \right )\right ) \mathit {Heaviside}\left (t -\frac {1}{3}\right )+\left (-3 \cos \left (2 \pi t \right ) \sqrt {3}-3 \sin \left (2 \pi t \right )\right ) \mathit {Dirac}\left (t -\frac {1}{3}\right )-2 \mathit {Dirac}\left (t -1\right ) \sin \left (2 \pi t \right )-4 \mathit {Heaviside}\left (t -1\right ) \pi \cos \left (2 \pi t \right )}{4 \pi } \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0 \\ {} & {} & 0=2 c_{2} \pi \\ {} & \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 {\left (-3 \cos \left (2 \pi t \right ) \sqrt {3}-3 \sin \left (2 \pi t \right )\right ) \mathit {Heaviside}\left (t -\frac {1}{3}\right )-2 \mathit {Heaviside}\left (t -1\right ) \sin \left (2 \pi t \right )}{4 \pi } \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {\left (-3 \cos \left (2 \pi t \right ) \sqrt {3}-3 \sin \left (2 \pi t \right )\right ) \mathit {Heaviside}\left (t -\frac {1}{3}\right )-2 \mathit {Heaviside}\left (t -1\right ) \sin \left (2 \pi t \right )}{4 \pi } \end {array} \]
Maple trace
✓ Solution by Maple
Time used: 5.843 (sec). Leaf size: 36
\[
y \left (t \right ) = \frac {\left (-3 \sqrt {3}\, \cos \left (2 \pi t \right )-3 \sin \left (2 \pi t \right )\right ) \operatorname {Heaviside}\left (t -\frac {1}{3}\right )-2 \sin \left (2 \pi t \right ) \operatorname {Heaviside}\left (t -1\right )}{4 \pi }
\]
✓ Solution by Mathematica
Time used: 0.125 (sec). Leaf size: 49
\[
y(t)\to -\frac {2 \theta (t-1) \sin (2 \pi t)+3 \theta (3 t-1) \left (\sin (2 \pi t)+\sqrt {3} \cos (2 \pi t)\right )}{4 \pi }
\]
4.30.2 Maple step by step solution
`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`
dsolve([diff(y(t),t$2)+(2*Pi)^2*y(t)=3*Dirac(t-1/3)-Dirac(t-1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
DSolve[{y''[t]+(2*Pi)^2*y[t]==3*DiracDelta[t-1/3]-DiracDelta[t-1],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]