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APPLICATIONS:

FIRST-ORDER DIFFERENTIAL EQUATION

GROWTH AND DECAY

In many applications, the rate of change of a variable y is proportional to the value of y. If y is a function of time t, the proportion can be written as follows:

𝑑𝑦

𝑑𝑡= 𝑘𝑦

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Theorem: Exponential Growth and Decay Model

If y is differentiable function of t such that 𝑦 > 0 𝑎𝑛𝑑 𝑦’ = 𝑘𝑦 for some constant k, then

𝑦 = 𝐶𝑒𝑘𝑡

C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decayoccurs when k < 0.

EXAMPLES:

1. The rate of change of y is proportional to y. When t = 0, y =2, and when t = 2, y =4. What is the value of y when t = 3?

2. Suppose that 10 grams of the plutonium isotope 239𝑃𝑢 was released in the Chernobyl nuclear accident. How long will it take for the 10 grams to decay to 1 gram? Half-life of plutonium is 24, 100 years.

3. Suppose an experimental population of fruit flies increases according to the law of exponential growth. There were 100 flies after the second day of the experiment and 300 flies after the fourth day. Approximately how many flies were in the original population?

4. Four months after it stops advertising, a manufacturing company notices that its sales have dropped from 100, 000 units per month to 80, 000 units per month. If the sales follow an exponential pattern of decline, what will they be after another 2 months?

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Newton’s Law of Cooling

States that the rate of change in the temperature of an object is proportional to the difference between the object’s temperature and the temperature of the surrounding medium.

Examples

1. Let y represent the temperature (in °F) of an object in a room whose temperature is kept at a constant 60°. If the object cools from 100° to 90° in 10 minutes, how much longer will it take for its temperature to decrease to 80°?

2. When an object is removed from a furnace and placed in an environment with a constant temperature of 80 deg Farenheit, its core temperature is 1500 deg F. One hour after it is removed, the core temperature is 1120 deg F. Find the core temperature 5 hours after the object is removed from the furnace

3. A container of hot liquid is placed in a freezer that is kept at a constant temperature of 20 deg F. The initial temperature of the liquid is 160 deg F. After 5 minutes, the liquid’s temperature is 60 deg F. How much longer will it take for its temperature to decrease to 30 deg F>

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Mixtures:Examples

1. A 200-gallon tank is half full of distilled water. At time t 0, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 5 gallons per minute, and the well-stirred mixture is withdrawn at the same rate. Find the amount Q of concentrate in the tank after 30 minutes.

2. A 100-gallon tank is full of a solution containing 25 pounds of a concentrate. Starting at time t = 0, distilled water is admitted to the tank at the rate of 5 gallons per minute, and the well-stirred solution is withdrawn at the same rate, as shown in the figure.

a. Find the amount Q of the concentrate in the solution as a function of time.

b. Find the time when the amount of concentrate in the tank reaches 15 pounds.

Consider a large tank holding 1000 L of pure water into which a brine solution of salt begins to flow at a constant rate of 6 L/min. The solution inside the tank is kept well stirred and is flowing out of the tank at a rate of 6 L/min. If the concentration of salt in the brine entering the tank is 0.1 kg/L, determine when the concentration of salt in the tank will reach 0.05 kg/L.

For the mixing problem described in the previous example, assume now that the brine leaves the tank at a rate of 5 L/min instead of 6 L/min, with all else being the same. Determine the concentration of salt in the tank as a function of time.

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Electrical Circuits

(a) RL circuit and (b) RC circuit

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Kirchoff’s Current Law: The algebraic sum of the currents flowing into any junction point must be zero.

Kirchoff’s Voltage Law. The algebraic sum of the instantaneous changes in potential (voltage drops) around any closed loop must be zero.

Ohm’s Law. The voltage drop ER across a resistor is proportional to the current I passing through the resistor.

Faraday’s law and Lenz’s Law. The voltage drop EL across an inductor is proportional to the instantaneous rate of change of the current I.

The voltage drop EC across capacitor is proportional to the electrical charge q on the capacitor.

Examples

1. An RL circuit with a 1-Ω resistor and a 0.01-H inductor is driven by a voltage E(t) = sin 100t V. If the initial inductor current is zero, determine the subsequent resistor and inductor voltages and the current.

2. An RC circuit with a 1- Ω resistor and a 0.000001-F capacitor is driven by a voltage E(t) = sin 100t V. If the initial capacitor voltage is zero, determine the subsequent resistor and capacitor voltages and the current.