Page 126 - Advanced engineering mathematics

106 CHAPTER 3 The Laplace Transform

5

4

3

2

1
0
0.5 1 1.5 2
t
–1
FIGURE 3.25 Graph of the output voltage in
Example 3.18.

SECTION 3.5 PROBLEMS

In each of Problems 1 through 5, solve the initial value does the position of this object compare with that of the
problem and graph the solution. object in Problem 6 at any positive time t?

1. y + 5y + 6y =3δ(t − 2) − 4δ(t − 5); y(0) = y (0) =0 8. A 2 pound weight is attached to the lower end of a
spring, stretching it 8/3 inches. The weight is allowed

2. y − 4y + 13y = 4δ(t − 3); y(0) = y (0) = 0

to come to rest in the equilibrium position. At some

3. y + 4y + 5y + 2y = 6δ(t); y(0) = y (0) = y (0) = 0 later time, which we call time 0, the weight is struck a

downward blow of magnitude 1/4 pound (an impulse).

4. y + 16y = 12δ(t − 5π/8); y(0) = 3, y (0) = 0
Assume no damping in the system. Determine the

5. y + 5y + 6y = Bδ(t); y(0) = 3, y (0) = 0 velocity with which the weight leaves the equilibrium

position as well as the frequency and magnitude of the
6. An object of mass m is attached to the lower end of a
oscillations.
spring of modulus k. Assume that there is no damping.
Derive and solve an equation of motion for the object, 9. Prove the filtering property of the delta function (The-
assuming that at time zero it is pushed down from the orem 3.6). Hint: Replace δ(t − a) with
equilibrium position with an initial velocity v 0 . With
what momentum does the object leave the equilibrium 1
position? lim (H(t − a − ) − H(t − a))
→0
7. Suppose, in the setting of Problem 6, the object is struck
a downward blow of magnitude mv 0 at time 0. How in the integral and interchange the limit and the integral.
3.6 Solution of Systems
Physical systems, such as circuits with multiple loops, may be modeled by systems of linear
differential equations. These are often solved using the Laplace transform (and later by matrix
methods).
We will illustrate the idea with a system having no particular significance, then look at a
problem in mechanics and one involving a circuit.
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