67.4.21 problem Problem 3(g)

Internal problem ID [14065]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 3(g)
Date solved : Tuesday, January 28, 2025 at 06:13:27 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+13 y&=39 \operatorname {Heaviside}\left (t \right )-507 \left (t -2\right ) \operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 11.127 (sec). Leaf size: 64

dsolve([diff(y(t),t$2)+4*diff(y(t),t)+13*y(t)=39*Heaviside(t)-507*(t-2)*Heaviside(t-2),y(0) = 3, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = 3-12 \operatorname {Heaviside}\left (t -2\right ) \left (\left (-\frac {5 \cos \left (6\right )}{12}+\sin \left (6\right )\right ) \sin \left (3 t \right )+\cos \left (3 t \right ) \left (\cos \left (6\right )+\frac {5 \sin \left (6\right )}{12}\right )\right ) {\mathrm e}^{-2 t +4}+3 \left (30-13 t \right ) \operatorname {Heaviside}\left (t -2\right )+\frac {{\mathrm e}^{-2 t} \sin \left (3 t \right )}{3} \]

✓ Solution by Mathematica

Time used: 0.057 (sec). Leaf size: 103

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+13*y[t]==39*UnitStep[t]-507*(t-2)*UnitStep[t-2],{y[0]==3,Derivative[1][y][0] ==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -39 t-12 e^{4-2 t} \cos (6-3 t)-5 e^{4-2 t} \sin (6-3 t)+\frac {1}{3} e^{-2 t} \sin (3 t)+93 & t>2 \\ \frac {1}{3} e^{-2 t} \sin (3 t)+3 & 0\leq t\leq 2 \\ \frac {1}{3} e^{-2 t} (9 \cos (3 t)+7 \sin (3 t)) & \text {True} \\ \end {array} \\ \end {array} \]