28.3.12 problem 6.47

Internal problem ID [4525]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 6. The Laplace Transform and Its Applications. Problems at page 291
Problem number : 6.47
Date solved : Monday, January 27, 2025 at 09:22:43 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=8 \left (t^{2}+t -1\right ) \operatorname {Heaviside}\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: 6.696 (sec). Leaf size: 52

dsolve([diff(y(t),t$2)+4*y(t)=8*(t^2+t-1)*Heaviside(t-2),y(0) = 1, D(y)(0) = 2],y(t), singsol=all)
 

\[ y = -9 \operatorname {Heaviside}\left (t -2\right ) \cos \left (2 t -4\right )-5 \operatorname {Heaviside}\left (t -2\right ) \sin \left (2 t -4\right )+\operatorname {Heaviside}\left (t -2\right ) \left (2 t^{2}+2 t -3\right )+\cos \left (2 t \right )+\sin \left (2 t \right ) \]

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 54

DSolve[{D[y[t],{t,2}]+4*y[t]==8*(t^2+t-1)*UnitStep[t-2],{y[0]==1,Derivative[1][y][0] == 2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \cos (2 t)+\sin (2 t) & t\leq 2 \\ 2 t^2+2 t-9 \cos (4-2 t)+\cos (2 t)+5 \sin (4-2 t)+\sin (2 t)-3 & \text {True} \\ \end {array} \\ \end {array} \]