Internal problem ID [847]
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: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y=\sin \left (t \right )-\operatorname {Heaviside}\left (t -2 \pi \right ) \sin \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.438 (sec). Leaf size: 25
dsolve([diff(y(t),t$2)+4*y(t)=sin(t)-Heaviside(t-2*Pi)*sin(t-2*Pi),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \frac {\sin \left (t \right ) \left (\cos \left (t \right )-1\right ) \left (-1+\operatorname {Heaviside}\left (t -2 \pi \right )\right )}{3} \]
✓ Solution by Mathematica
Time used: 0.061 (sec). Leaf size: 27
DSolve[{y''[t]+4*y[t]==Sin[t]-UnitStep[t-2*Pi]*Sin[t-2*Pi],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {2}{3} \theta (2 \pi -t) \sin ^2\left (\frac {t}{2}\right ) \sin (t) \]