Internal problem ID [6684]
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. 7.3.1 TRANSLATION ON THE s-AXIS.
Page 297
Problem number: 67.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y=\sin \left (t \right ) \operatorname {Heaviside}\left (t -2 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 2.031 (sec). Leaf size: 28
dsolve([diff(y(t),t$2)+4*y(t)=sin(t)*Heaviside(t-2*Pi),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = -\frac {\left (\cos \left (t \right )-1\right ) \sin \left (t \right ) \operatorname {Heaviside}\left (t -2 \pi \right )}{3}+2 \cos \left (t \right )^{2}-1 \]
✓ Solution by Mathematica
Time used: 0.065 (sec). Leaf size: 36
DSolve[{y''[t]+4*y[t]==Sin[t]*UnitStep[t-2*Pi],{y[0]==1,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \cos (2 t) & t\leq 2 \pi \\ \frac {1}{3} (3 \cos (2 t)-\cos (t) \sin (t)+\sin (t)) & \text {True} \\ \end {array} \\ \end {array} \]