Internal problem ID [852]
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.
CAS Maple gives this as 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*}
✓ Solution by Maple
Time used: 0.328 (sec). Leaf size: 23
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)
\[ y \left (t \right ) = -\frac {\left (\cos \left (t \right )+1\right )^{2} \operatorname {Heaviside}\left (t -\pi \right )}{6}+\frac {\left (\cos \left (t \right )-1\right )^{2}}{6} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 29
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]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {2}{3} \sin ^4\left (\frac {t}{2}\right ) & t\leq \pi \\ -\frac {2 \cos (t)}{3} & \text {True} \\ \end {array} \\ \end {array} \]