Chapter 7



Chapter 7

Modeling and Analysis of Real World Systems by Higher Order

Differential Equations

1. Series Electrical circuit

2. Falling Bodies

3. The shape of a Hanging cable. The power line problem

4. Diabetes and glucose Tolerance Test

5. Rocket Motion

6. Undamped and Damped motion

7. Exercises

In this chapter we discuss solutions of second order linear differential equations representing LCR series electrical circuit, falling bodies with or without resistance, the shape of a hanging cable, diabetes and glucose tolerance test, curve of pursuit, rocket motion, damped and free damped motion.

7.1 Series Electrical Circuit

Every electrical circuit has three passive elements, called inductor, resistor, and capacitor that behave to impede the flow of current, the change in current, or the change in voltage. These three constants are denoted respectively by L,R and C.

If i(t) denotes current in the LRC series electrical circuit shown in Figure 7.1 then the voltage drops across the inductor, resistor, and capacitor are [pic] By Kirchhoff’s second law, the sum of these voltages equals the voltage E (t) impressed on the circuit; that is,

L [pic] + Ri + [pic]q = E(t) (7.1)

But the charge q(t) on the capacitor is related to the current i(t) by i = [pic], and so (7.1) becomes the linear second-order differential equation

L[pic]+ R[pic] + [pic]q (t) = E (t) (7.2)

Figure 7.1 LCR Series Circuit

If E (t) = 0, the electrical vibrations of the circuit are said to be free. Since the auxiliary equation (7.2) is Lm2+ Rm + [pic] = 0 (7.3)

there will be three forms of the solution with R(0, depending on the value of the discriminant R2- [pic] The circuit is called

i) Overdamped if R2 - [pic] >0

ii) Critically damped if R2-[pic]= 0

iii) Underdamped if R2-[pic] 0 then the general solution of (7.2) is of the form

y(x)=c1em1x+c2em2x (See equation (5.16)).

Case (ii) If R2- [pic] then m1=m2 and the general solution is of the form y(x)=c1em1x+c2xem1x (See equation (5.17)).

Case (iii) If R2-[pic] 0 if inductance L=1, Resistance R=300, capacitance C=5x10-5 and at t=0 the switch is closed to a 40-volt battery. Furthermore it is assumed that q(0) = i(0)=0.

Solution : This LRC electrical circuit is represented by (7.2)

where L=1, R=300, C=5x10-5, E(t)=40

and so q(t) is the general solution of the nonhomogeneous linear differential equation with constant coefficients.

[pic]+ 300 [pic]+ [pic]q= 40

subject to q(0)= i(0)=0

The auxiliary equation is

m2+300m+20000=0

having roots

m1= - 100, m2 = - 200

Since g(t) is constant, a particular solution is of the form qp = A so

20000 A=40 or A=2x10-3=0.002

(Apply the method of undetermined coefficients for finding a particular solution).

q(t) = c1 e -100t+c2e-200t+0.002

q(0)=c1e-100x0 +c2e-200x0+0.002

0= c1+c2+0.002

i(t) = [pic]= - 100c1e-100t-200c2e-200t

0=i(0)= -100c1-200c2 or c1+2c2=0

Therefore c1= - 0.004 and c2 = 0.002.

This gives q(t) = 0.002(-2e-100t+e-200t+1)

i(t) = 0.4 (e-100t-e-200t).

Example 7.2 Find the charge on the capacitor in an LRC series circuit at t=0.01s when L=0.05 henry , R=2(, C=0.01F, E(t)=0 V, q(0)=5 coulomb, and i(0)=0. Determine the first time at which the charge on the capacitor is equal to zero.

Solution: The model for LRC series circuit is

[pic] + [pic]+ [pic]q(t) = E(t)

where L= .05, R=2, C=.01, E(t)= 0,

and i(0)=0. The auxiliary equation is

[pic]m2+2m+100=0

or m2+40m+2000=0

Roots are m = [pic]

m1= -20+i40, that is, (= - 20, (= 40

m2= -20-i40

In view of equation (5.18)

q(t)=e-20t (c1 cos 40t +c2 sin 40 t)

The initial conditions q(0)=5 and q'(0)=0

imply 5=e-0(c1cos 0 +c2 sin 0)

or c1= 5 and

q'(t) = - 20e-20t(c1cos 40t +c2 sin 40t)+e-20t(-40c1 sin 40t + 40c2 cos 40t)

q'(0)=0= -20(c1)+1 (40c2)=0

or -c1+2c2=0. Since c1=5 we have

2c2=5 or c2= [pic].

Thus q(t)=e-20t(5 cos 40t + [pic]sin 40t)

( [pic]e-20t sin (40t +1.1071)

and q (0.01) ( 4.5676 Columbus.

The charge is zero for the first time when

40t+1.1071 = ( or t ( 0.0509 seconds.

7.2 Falling Bodies

The motion of a falling body (say, a rock) without any resistance except the force of gravity is described by the equation

[pic] [pic]= -mg

or [pic]= - g, (7.4)

where s(t) denotes the position of the rock relative to the ground at time t, and g is the acceleration due to gravity. The minus sign is used in (7.4) as the weight of the rock is a force directed downwards, which is opposite to the positive direction. If the height of the place from where rock is tossed is s0 and the initial velocity of the rock is vo, then s is determined from the second order initial-value problem

[pic]= -g, s(0)=so, s'(0)=vo (7.5)

Integrating twice (7.4) and using the initial condition of (7.5) we get

s(t) = -[pic] gt2 + vo t + so (7.6)

In model (7.4) we have ignored air resistance, However, in many cases it is quite significant such as a feather with low density and irregular shape encounters air resistance proportional to its instantaneous velocity v. It we take, in this circumstance, the positive direction to be oriented downwards, then the net force acting on the object is given by mg-kv. In this case motion of the falling body is modelled by

[pic]+ k [pic]= mg, (7.7)

where v=[pic], m is the mass of the falling body and g is the acceleration due to gravity.

Example 7.3

Suppose a Cannon ball weighing 16 pounds is shot vertically upwards with an initial velocity v0 = 300 ft/s. Find a) the velocity at any time t. b) the maximum height attained by the cannon ball. (ignore air resistance)

Solution: a) Taking the positive direction as upwards, the required equation is

[pic]

Integrating, we get [pic].

From v(0) = 300, we find c = 300, so the velocity is v(t) = -32t + 300.

b) Integrating again and using s(0) = 0, we get

s(t) = - 16 t2 + 300t.

The maximum height is attained when v=0 that is at t=9.375.

The maximum height is s (9.375) = 1406.25 ft

7.3 The shape a Hanging Cable. The power line problem

Figure 7.2(a) Figure 7.2(b)

Suppose a flexible cable is hung from two points A and B as shown in Figure 7.2(a) & 7.2(b). Assume that the vertical loading (its weight and any external forces) cause the cable to take the shape shown in the figure 7.2b with its lowest point P1 and any arbitrary point P2. Three forces are acting on the wire namely, weight of the segment P1P2 and tension T1 andT2 in the wire at P1 and P2, respectively. If w is the linear density of the wire (measured, say, in kg/meter) and s is the length of the segment P1P2 then its weight is ws. We resolve T2 into vertical and horizontal components; T2 sin ( and T2 cos ( are respectively vertical and horizontal components. Thus

T2cos (=T1 and T2 sin ( = ws

These two equations imply that

tan ( = [pic] that is

[pic]= [pic]

Differentiating this equation with respect to x we get

[pic]= [pic] [pic]

Since the arc length between P1 and P2 is given by

s = [pic][pic]

[pic] = [pic]

Therefore

[pic] = [pic][pic] (7.8)

The solution of (7.8) will provide the shape of a hanging cable.

Example 7.4 Find the solution of the initial-value problem

[pic] = [pic] [pic]

y(0)= [pic], y'(0)=0

Solution: If we substitute u=y' then the given differential equation takes the form

[pic]=[pic](1+u2)[pic]

Separating variables, we get

[pic] = [pic]dx

By integrating both sides we obtain

[pic][pic] = [pic][pic]

sinh-1u = [pic] x + c1

y’ (0) = 0 is equivalent to u(0) = 0.

Thus

sin h-10 = [pic] 0 + c1

or c1 = sinh-10 = 0.

This implies that

u = sinh [pic] x

or [pic] = sinh [pic]

Integrating both sides we get

y = [pic]cosh [pic] + c2

By initial condition

y(0) = [pic], we obtain

[pic]= [pic]+ c2 or c2=0

Therefore y=[pic] cosh [pic] (7.9)

The curve given by (7.9) is called a catenary. The shape of a cable is a catenary under appropriate initial conditions. The word catenary has been derived from Latin word catena meaning chain. Shape of transmission lines, telegraph cables and cables of suspension bridges are of the catenary form.

7.4 Diabetes and Glucose Tolerance Test

Percentage of sugar in the blood is an important factor in the metabolism of human body. Excessive deviation of percentage of sugar from normal concentration leads to severe dysfunction and causes deadly diseases leading to death. Diabetes mellitus is a disease which is characterized by percentage of sugar deviating from the normal value. Diabetes is diagonized by means of a glucose tolerance test (GTT). The following differential equation of order 2 in quite relevant in this problem.

[pic] + 2 ( [pic]+ w02g = 0 (7.10)

where g = G – Go, H=H-Ho,

G and H denote respectively the sugar concentration and the net effective hormonal concentration. Go and Ho denote respectively the equilibrium values of G and H at the beginning of the GTT. ( and w0 are appropriate constants.

Studies of normal and mildly diabetic persons showed that (2- wo2 ................
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