Internal problem ID [13236]
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 6.6. page 624
Problem number: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+y^{\prime }+5 y=\operatorname {Heaviside}\left (-2+t \right ) \sin \left (-8+4 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 6.703 (sec). Leaf size: 89
dsolve([diff(y(t),t$2)+diff(y(t),t)+5*y(t)=Heaviside(t-2)*sin(4*(t-2)),y(0) = -2, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \frac {4 \cos \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{1-\frac {t}{2}}}{137}+\frac {92 \sin \left (\frac {\sqrt {19}\, \left (t -2\right )}{2}\right ) \operatorname {Heaviside}\left (t -2\right ) \sqrt {19}\, {\mathrm e}^{1-\frac {t}{2}}}{2603}-2 \,{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {19}\, t}{2}\right )-\frac {2 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {19}\, \sin \left (\frac {\sqrt {19}\, t}{2}\right )}{19}-\frac {4 \left (\cos \left (4 t -8\right )+\frac {11 \sin \left (4 t -8\right )}{4}\right ) \operatorname {Heaviside}\left (t -2\right )}{137} \]
✓ Solution by Mathematica
Time used: 6.103 (sec). Leaf size: 163
DSolve[{y''[t]+y'[t]+5*y[t]==UnitStep[t-2]*Sin[4*(t-2)],{y[0]==-2,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -\frac {2}{19} e^{-t/2} \left (19 \cos \left (\frac {\sqrt {19} t}{2}\right )+\sqrt {19} \sin \left (\frac {\sqrt {19} t}{2}\right )\right ) & t\leq 2 \\ \frac {e^{-t/2} \left (-76 e^{t/2} \cos (8-4 t)+76 e \cos \left (\frac {1}{2} \sqrt {19} (t-2)\right )-5206 \cos \left (\frac {\sqrt {19} t}{2}\right )+209 e^{t/2} \sin (8-4 t)+92 \sqrt {19} e \sin \left (\frac {1}{2} \sqrt {19} (t-2)\right )-274 \sqrt {19} \sin \left (\frac {\sqrt {19} t}{2}\right )\right )}{2603} & \text {True} \\ \end {array} \\ \end {array} \]