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# Solution to first and second order differential equations

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Question 5
Suppose Anytown, USA has a fixed population of 200,000. On
March 1, 3000 people have the flu. On June 1, 6000 people have it.
If the rate of increase of the number N(t) who have the flu is
proportional to the number who don't have it, how many will have
the disease on September 1?

Question 7
Suppose that a motorboat is moving at 30 ft/sec when its motor
suddenly quits, and that 5 seconds later the boat has slowed to 15
ft/sec. Assume that the resistance it encounters while coasting is
proportional to its velocity. How far will the boat coast in all?

Question 12
A mass of 25g is attached to a vertical spring with spring constant
k = 3 dyne/cm. The surrounding medium has a damping constant of
10 dyne*sec/cm. The mass is pushed 5 cm above its equilibrium
position and released. Find (a) the position function of the mass,
(b) the period of the vibration, and (c) the frequency of the vibration.

##### Solution Summary

I have solved 3 word problems. Each of them involves the solution to first and second order differential equations. I have constructed the differential equations using physics principles and solved them using the given boundary conditions.

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Question 5
Suppose Anytown, USA has a fixed population of 200,000. On
March 1, 3000 people have the flu. On June 1, 6000 people have it.
If the rate of increase of the number N(t) who have the flu is
proportional to the number who don't have it, how many will have
the disease on September 1?
dN/dt = k (p - N) where p = 200, 000 and k is a constant.
Let t =0 is march , t = 3 is june and t = 6 is sep.
dN/dt + kN = kp
N(t) = p + A e^(-kt) ----------- (1)

t = 0 , N = 3000
t = 3, N = 6000
3000 = 200, 000 + A e^(-k*3) è A = -197, 000
6000 = 200, 000 - 197, 000 * e^(-3k) => ...

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