MATH 225 - Differential Equations June 9 , 2008
Homework 8, Field 2008 Due Date: June 12, 2008
Laplace Transforms - Second-Order Linear Equations (revisited) - Forced Mass-Spring Systems
1. Calculate the Laplace transform of f1(t) = sinh(at) and f2(t) = cosh(at), a ∈R.
Hint: You may want to refer to the previous homework assignment for the definitions of sinh(x) and cosh(x)
or you might find it more efficient to note that −isin(ix) = sinh(x) and cos(ix) = cosh(x) and repeat the
calculations we did in class to find L{cos(kt)}and L{sin(kt)}, which will find the transform of f1and f2
simultaneously. Doing it this way should make the standard form for these transforms, http://en.wikipedia.
org/wiki/Laplace_transform#Table_of_selected_Laplace_transforms, make more sense.
2. Using the definition of transform show the following relationships:
(a) L{eatf(t)}=F(s−a)
(b) L{f(t−a)ua(t)}=e−asF(s), a ≥0
(c) L{f(t)ua(t)}=e−asL{f(t+a)}, a ≥0.
3. Consider the following second-order linear ordinary differential equation with constant coefficients,
ad2y
dt2+bdy
dt +cy =δ(t), y(0) = 0, y0(0) = 0.(1)
Solve the IVP (1) for the following cases:
(a) a= 1, b=−2, c=−3
(b) a= 1, b= 4, c= 4
(c) a= 1, b=−4, c= 13
(d) a= 1, b= 0, c= 9
4. Given the following forced simple harmonic oscillator.
2d2y
dt2+ 8y= 6 cos(ωt), y(0) = 1, y0(0) = −1.(2)
(a) Set ω= 1 and find the solution to the initial value problem.
(b) Set ω= 2 and find the solution to the initial value problem.
(c) Describe the differences in the long term behavior of the steady-state solution for each oscillator
5. Again we investigate the forced mass spring system given by,
md2y
dt2+bdy
dt +ky =f(t), m, b, k ∈R+∪ {0}.(3)
(a) Suppose we have that b= 0, y(0) = α,y0(0) = βand f(t) = AδT(t), show that the solution to (3) subject
to these constraints is given by,
y(t) = αcos(ωt) + β
ωsin(ωt) + A
mω uT(t) sin(ω(t−T)),(4)
where ω2=k
m.
(b) Suppose that we wish to hit the mass in such a way that after the impact the oscillations stop. Show that
for this to occur we must choose,
A=αmω
sin(ωT )(5)
T=1
ωarctan −αω
β.(6)
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