Internal problem ID [852]
Internal file name [OUTPUT/852_Sunday_June_05_2022_01_51_37_AM_29676591/index.tex]
Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima,
Meade
Section: Chapter 6.4, The Laplace Transform. Differential equations with discontinuous forcing
functions. page 268
Problem number: 8.
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "higher_order_laplace"
Maple gives the following as the ode type
[[_high_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime \prime }+5 y^{\prime \prime }+4 y=1-\operatorname {Heaviside}\left (t -\pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 0] \end {align*}
Solving using the Laplace transform method. Let \[ \mathcal {L}\left (y\right ) =Y(s) \] 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 )\\ \mathcal {L}\left (y^{\prime \prime \prime }\right ) &= s^3 Y(s) - y''(0) - s y'(0) - s^2 y \left (0\right )\\ \mathcal {L}\left (y^{\prime \prime \prime \prime }\right ) &= s^4 Y(s) - y'''(0) - s y''(0) - s^2 y'(0)- s^3 y \left (0\right ) \end {align*}
The given ode becomes an algebraic equation in the Laplace domain \[ s^{4} Y \left (s \right )-y^{\prime \prime \prime }\left (0\right )-s y^{\prime \prime }\left (0\right )-s^{2} y^{\prime }\left (0\right )-s^{3} y \left (0\right )+5 s^{2} Y \left (s \right )-5 y^{\prime }\left (0\right )-5 s y \left (0\right )+4 Y \left (s \right ) = \frac {1-{\mathrm e}^{-s \pi }}{s}\tag {1} \] But the initial conditions are \begin {align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=0 \end {align*}
Substituting these initial conditions in above in Eq (1) gives \[ s^{4} Y \left (s \right )+5 s^{2} Y \left (s \right )+4 Y \left (s \right ) = \frac {1-{\mathrm e}^{-s \pi }}{s} \] Solving the above equation for \(Y(s)\) results in \[ Y(s) = -\frac {-1+{\mathrm e}^{-s \pi }}{s \left (s^{4}+5 s^{2}+4\right )} \] Taking the inverse Laplace transform gives \begin {align*} y&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (-\frac {-1+{\mathrm e}^{-s \pi }}{s \left (s^{4}+5 s^{2}+4\right )}\right )\\ &= -\frac {2 \cos \left (t \right )}{3}+\frac {\operatorname {Heaviside}\left (\pi -t \right ) \left (\cos \left (t \right )+1\right )^{2}}{6} \end {align*}
Converting the above solution to piecewise it becomes \[
y = \left \{\begin {array}{cc} -\frac {2 \cos \left (t \right )}{3}+\frac {\left (\cos \left (t \right )+1\right )^{2}}{6} & t \le \pi \\ -\frac {2 \cos \left (t \right )}{3} & \pi The solution(s) found are the following \begin{align*}
\tag{1} y &= \left \{\begin {array}{cc} -\frac {2 \cos \left (t \right )}{3}+\frac {\left (\cos \left (t \right )+1\right )^{2}}{6} & t \le \pi \\ -\frac {2 \cos \left (t \right )}{3} & \pi Verification of solutions
\[
y = \left \{\begin {array}{cc} -\frac {2 \cos \left (t \right )}{3}+\frac {\left (\cos \left (t \right )+1\right )^{2}}{6} & t \le \pi \\ -\frac {2 \cos \left (t \right )}{3} & \pi Maple trace
✓ Solution by Maple
Time used: 0.328 (sec). Leaf size: 23
\[
y \left (t \right ) = -\frac {\left (\cos \left (t \right )+1\right )^{2} \operatorname {Heaviside}\left (t -\pi \right )}{6}+\frac {\left (\cos \left (t \right )-1\right )^{2}}{6}
\]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 29
\[
y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {2}{3} \sin ^4\left (\frac {t}{2}\right ) & t\leq \pi \\ -\frac {2 \cos (t)}{3} & \text {True} \\ \end {array} \\ \end {array}
\]
`Methods for high order ODEs:
--- Trying classification methods ---
trying a quadrature
trying high order exact linear fully integrable
trying differential order: 4; linear nonhomogeneous with symmetry [0,1]
trying high order linear exact nonhomogeneous
trying differential order: 4; missing the dependent variable
checking if the LODE has constant coefficients
<- constant coefficients successful`
dsolve([diff(y(t),t$4)+5*diff(y(t),t$2)+4*y(t)=1-Heaviside(t-Pi),y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(t), singsol=all)
DSolve[{y''''[t]+5*y''[t]+4*y[t]==1-UnitStep[t-Pi],{y[0]==0,y'[0]==0,y''[0]==0,y'''[0]==0}},y[t],t,IncludeSingularSolutions -> True]