Internal problem ID [12908]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 6. Laplace transform. Section 6.3 page 600
Problem number: 32.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+3 y=\operatorname {Heaviside}\left (t -4\right ) \cos \left (5 t -20\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -2] \end {align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 39
dsolve([diff(y(t),t$2)+3*y(t)=Heaviside(t-4)*cos(5*(t-4)),y(0) = 0, D(y)(0) = -2],y(t), singsol=all)
\[ y = -\frac {2 \sqrt {3}\, \sin \left (\sqrt {3}\, t \right )}{3}-\frac {\operatorname {Heaviside}\left (t -4\right ) \cos \left (5 t -20\right )}{22}+\frac {\operatorname {Heaviside}\left (t -4\right ) \cos \left (\sqrt {3}\, \left (t -4\right )\right )}{22} \]
✓ Solution by Mathematica
Time used: 0.797 (sec). Leaf size: 66
DSolve[{y''[t]+3*y[t]==UnitStep[t-4]*Cos[5*(t-4)],{y[0]==0,y'[0]==-2}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -\frac {2 \sin \left (\sqrt {3} t\right )}{\sqrt {3}} & t\leq 4 \\ \frac {1}{66} \left (-3 \cos (5 (t-4))+3 \cos \left (\sqrt {3} (t-4)\right )-44 \sqrt {3} \sin \left (\sqrt {3} t\right )\right ) & \text {True} \\ \end {array} \\ \end {array} \]