72.19.7 problem 33

Internal problem ID [14965]
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
Date solved : Tuesday, January 28, 2025 at 07:25:42 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+9 y&=20 \operatorname {Heaviside}\left (t -2\right ) \sin \left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 12.850 (sec). Leaf size: 67

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)
 

\[ y = {\mathrm e}^{-2 t +4} \cos \left (\sqrt {5}\, \left (t -2\right )\right ) \operatorname {Heaviside}\left (t -2\right )+{\mathrm e}^{-2 t} \cos \left (t \sqrt {5}\right )+\frac {4 \,{\mathrm e}^{-2 t} \sqrt {5}\, \sin \left (t \sqrt {5}\right )}{5}-\operatorname {Heaviside}\left (t -2\right ) \left (\cos \left (t -2\right )-2 \sin \left (t -2\right )\right ) \]

✓ Solution by Mathematica

Time used: 1.395 (sec). Leaf size: 115

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+9*y[t]==20*UnitStep[t-2]*Sin[t-2],{y[0]==1,Derivative[1][y][0] ==2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -\cos (2-t)+e^{4-2 t} \cos \left (\sqrt {5} (t-2)\right )+e^{-2 t} \cos \left (\sqrt {5} t\right )-2 \sin (2-t)+\frac {4 e^{-2 t} \sin \left (\sqrt {5} t\right )}{\sqrt {5}} & t>2 \\ \frac {1}{5} e^{-2 t} \left (5 \cos \left (\sqrt {5} t\right )+4 \sqrt {5} \sin \left (\sqrt {5} t\right )\right ) & \text {True} \\ \end {array} \\ \end {array} \]