1. Differential Equations

2. An equation involving a first derivative is called a first-order differential equation. Some examples are:

3. An equation involving a second derivative is called a second-order differential equation. Some examples are:

4. In this class, we will look only at first-order differential equations.
We can solve these equations by integrating.
Integrating a differential equation will provide either a general or particular solution.
There are many different methods of solving a differential equation. In this class, we will learn a method call “separation of variables.”

5. Example:
Example:

6. Find the particular solution of the following differential equation:
Answer:
Find the particular solution of the following differential equation:
Answer:

7. Find the particular solution of the following differential equation:
Answer:
Find the particular solution of the following differential equation:
Answer:

8. Find the particular solution of the following differential equation:
Answer:
Find the general solution of the following differential equation:
Answer:

9. Consider the population of a country
Consider the population of a country. If there is no immigration or emigration, the rate at which the population is changing is often proportional to the population. For example, if the population at time t is P and its continuous growth rate is 2%, then:

10. Newton’s Law of Cooling
Newton proposed that the temperature of a hot object decreases at a rate proportional to the difference between its temperature and that of its surroundings. Likewise, a cold object heats up at a rate proportional to the temperature difference between the object ad its surroundings.
Therefore, we can model a problem using an exponential function which approaches an asymptote that is the room temperature.

11. Example:
When a murder is committed, the body, originally at 37 degrees, cools according to Newton’s Law of Cooling. Suppose that after two hours the temperature of the body is 35 degrees and that the temperature of the surrounding air is 20 degrees. Find the temperature H of the body as a function of time t, the time in hours since the murder was committed. If the body is found at 4:00 PM at a temperature of 30 degrees, when was the murder committed?
We know that H = 37, when t = 0
the murder was committed about 8.43 hours before 4:00 PM
We know that H = 35 when t = 2

12. At time t minutes, the rate of change of temperature of a cooling body is proportional to the temperature T of that body at that time. Initially T = 72 degrees Celsius. Given that T = 32 degrees when t = 10, find a function relating T to t and find out how much longer it will take the body to cool to 27 degrees under these conditions.
Answer: