1. Introductory Concepts
Chapter 1
Introductory Concepts

2. Sec 1.2 – Basic Concepts
A differential equation is an equation that relates an unknown function and one or more of its derivatives
So if y = f(x), we can write for example
Usually y is the function with x its independent variable (motion in the xy-plane), or x is the function with t its independent variable (particle motion along the x-axis with respect to time)

3. Examples .

4. Solutions of DE’s A solution of a DE is a function y(x) (or x(t)) that
is continuous on an interval on the x-axis (t-axis)
Satisfies the equation in question
The interval can be found, usually by looking at the equation and seeing where everything’s defined.

5. Solutions of DE’s Some solutions can be found just by inspection
Example: Find a solution y(x) to the equation

6. Solutions of DE’s Some are tougher, but can be verified
Example: Verify that is a solution to the differential equation

7. Notation and Terminology
The order of a DE is the highest derivative of the solution function that appears.
The general form of an nth-order differential with unknown function y of independent variable x is
So there can be terms involving x (the independent variable), y (the function itself), y’ (its first derivative), and so on
You can now try problems 1 and 2 in Sec 1.2

8. You’ve already seen some DE in Calc
“The rate of change of a population is proportional to its size”
Find functions for the velocity and altitude of a projectile fired upward from ground level with initial velocity 100 ft/sec.

9. Integrals; Particular Solutions
Suppose that the derivative of the function y(x) that we seek depends only on x’s, and not on y.
IOW,
Then

10. Examples 1.
Then =
For what interval on the x-axis is this solution valid?
So there are an infinite number of solutions.
How can we find a particular one?

11. Example, with initial value
Suppose we know that
And that when x=1, y=3. We can now find C, and write a particular solution.

12. Another example of an initial value problem
Find a solution to the IVP

13. 2nd Order Equation General: Then y’= So y=
And we need two initial conditions to find

14. 2nd Order Example
Ex:
Then

15. Common 2nd-order DE– a(t), v(t), x(t)
As we know,
Thus
As we have seen, in order to find a particular x(t), we will need an initial velocity and an initial position
In the particular case where the acceleration is constant a,
After that, just need to line up the units and ID the question(s) being asked.

16. Example
A ball is thrown straight down from a building w initial speed of 10 m/sec. It hits with speed 60 m/sec. How tall is the building?

17. Example
Find y, if
Now do the rest of the Sec 1.2 problems on the schedule

18. Example
Find y, if y=
Let
Then