Internal problem ID [12909]
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: 33.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+9 y=20 \operatorname {Heaviside}\left (-2+t \right ) \sin \left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 64
dsolve([diff(y(t),t$2)+4*diff(y(t),t)+9*y(t)=20*Heaviside(t-2)*sin(t-2),y(0) = 1, D(y)(0) = 2],y(t), singsol=all)
\[ y = {\mathrm e}^{4-2 t} \cos \left (\sqrt {5}\, \left (-2+t \right )\right ) \operatorname {Heaviside}\left (-2+t \right )+{\mathrm e}^{-2 t} \cos \left (\sqrt {5}\, t \right )+\frac {4 \sqrt {5}\, {\mathrm e}^{-2 t} \sin \left (\sqrt {5}\, t \right )}{5}-\operatorname {Heaviside}\left (-2+t \right ) \left (\cos \left (-2+t \right )-2 \sin \left (-2+t \right )\right ) \]
✓ Solution by Mathematica
Time used: 2.391 (sec). Leaf size: 115
DSolve[{y''[t]+4*y'[t]+9*y[t]==20*UnitStep[t-2]*Sin[t-2],{y[0]==1,y'[0]==2}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -\cos (2-t)+e^{4-2 t} \cos \left (\sqrt {5} (t-2)\right )+e^{-2 t} \cos \left (\sqrt {5} t\right )-2 \sin (2-t)+\frac {4 e^{-2 t} \sin \left (\sqrt {5} t\right )}{\sqrt {5}} & t>2 \\ \frac {1}{5} e^{-2 t} \left (5 \cos \left (\sqrt {5} t\right )+4 \sqrt {5} \sin \left (\sqrt {5} t\right )\right ) & \text {True} \\ \end {array} \\ \end {array} \]