72.19.6 problem 32

Internal problem ID [14964]
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
Date solved : Tuesday, January 28, 2025 at 07:25:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y&=\operatorname {Heaviside}\left (t -4\right ) \cos \left (5 t -20\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 11.584 (sec). Leaf size: 42

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)
 

\[ y = \frac {\operatorname {Heaviside}\left (t -4\right ) \cos \left (\sqrt {3}\, \left (t -4\right )\right )}{22}-\frac {2 \sqrt {3}\, \sin \left (\sqrt {3}\, t \right )}{3}-\frac {\operatorname {Heaviside}\left (t -4\right ) \cos \left (5 t -20\right )}{22} \]

✓ Solution by Mathematica

Time used: 1.481 (sec). Leaf size: 202

DSolve[{D[y[t],{t,2}]+3*y[t]==UnitStep[t-4]*Cos[5*(t-4)],{y[0]==0,Derivative[1][y][0] ==-2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {i e^{-i \sqrt {3} t} \left (-1+e^{2 i \sqrt {3} t}\right )}{\sqrt {3}} & t\leq 4 \\ \frac {1}{132} e^{-i \left (5+\sqrt {3}\right ) (t+4)} \left (\cos \left (\sqrt {3} t\right )+i \sin \left (\sqrt {3} t\right )\right ) \left (-3 e^{4 i \sqrt {3}} \left (e^{40 i}+e^{10 i t}\right )+3 e^{5 i (t+4)} \left (1+e^{8 i \sqrt {3}}\right ) \cos \left (\sqrt {3} t\right )-e^{5 i (t+4)} \left (-3 i+88 \sqrt {3} e^{4 i \sqrt {3}}+3 i e^{8 i \sqrt {3}}\right ) \sin \left (\sqrt {3} t\right )\right ) & \text {True} \\ \end {array} \\ \end {array} \]