2. Differential Equations Definition A differential equation is an equation involving derivatives of an unknown function and possibly the function itself as well as the independent variable. Example Definition The order of a differential equation is the highest order of the derivatives of the unknown function appearing in the equation 1 st order equations2 nd order equation Examples In the simplest cases, equations may be solved by direct integration. Observe that the set of solutions to the above 1 st order equation has 1 parameter, while the solutions to the above 2 nd order equation depend on two parameters.

3. Separable Differential Equations A separable differential equation can be expressed as the product of a function of x and a function of y. Example: separate the variables. (Assume y 2 is never zero.) Combined constants of integration

4. In many natural phenomena, quantities grow or decay at a rate proportional to their size. The number of bighorn sheep in a population increases at a rate that is proportional to the number of sheep present (at least for awhile.) So does any population of living creatures. Other things that increase or decrease at a rate proportional to the amount present include radioactive material and money in an interest-bearing account.

5. So if the rate of change is proportional to the amount present, the change can be modeled by:

6. k is either a growth or decay constant k is positive: equation represents growth. k is negative: equation represents decay. Law of Exponential Change – Law of Natural Growth or Decay Law of Exponential Change – Law of Natural Growth or Decay

7. What is the significance of the proportionality constant k? In the context of population growth, where P(t) is the size of a population at time t, we can write: It is called the relative growth rate instead of saying “the growth rate is proportional to population size” we could say “the relative growth rate is constant.” Population with constant relative growth rate must grow exponentially.

8. Example 1 Bacteria in a culture increased from 400 to 1600 in three hours. Assuming that the rate of increase is directly proportional to the population, a) Find an appropriate equation to model the population. b) Find the number of bacteria at the end of six hours. If we were to solve for k rather than e k we might have the problem with rounding!!

9. Continuously Compounded Interest If money is invested in a fixed-interest account where the interest is added to the account r times per year, the amount present after t years is: If the money is added back more frequently, you will make a little more money. The best you can do is if the interest is added continuously. Of course, the bank does not employ some clerk to continuously calculate your interest with an adding machine. We could calculate if we only knew how (we’ll know) We peek:

10. Example 3 Find the amount of money in a bank account with a 4% interest rate after 10 years if originally there was $5000 in it. k = 0.04 P 0 = 5000 t = 10 years

11. The equation for the amount of a radioactive element left after time t is: This allows the decay constant, k, to be positive. In problems: The half-life is the time required for half the material to decay. Radioactive Decay k is not a half-life

12. Example 2 Carbon-14 has a half life of approximately 5730 years (every 5730 years, the amount of radioactive substance will be halved). It is often used in carbon dating to find the age of artifacts and fossils since the amount of carbon-14 in the atmosphere is known. Assume that a certain fossil has 30% as much carbon-14 as its present-day equivalent should have. Approximate the age of the fossil. take 5730 th root by the way, k = – 0.000120968 exponential decay

13. Espresso left in a cup will cool to the temperature of the surrounding air. The rate of cooling is proportional to the difference in temperature between the liquid and the air. (It is assumed that the air temperature is constant.) If we solve the differential equation:we get: Newton’s Law of Cooling Newton's Law makes a statement about an instantaneous rate of change of the temperature. Newton’s Law of Cooling

14. Example 4 A pot of liquid is put on the stove to boil. The temperature of the liquid reaches 170 o F and then the pot is taken off the burner and placed on a counter in the kitchen. The temperature of the air in the kitchen is 76 o F. After two minutes the temperature of the liquid in the pot is 123 o F. How long before the temperature of the liquid in the pot will be 84 o F?

15. The other way from here sometimes much bigger difference!!!!!!!!!!