1. DIFFERENTIAL EQUATIONS

2. What is a differential equation?
A differential equation is any equation that contains a derivative
First Order differential equations
Second Order differential equations

3. What will we do with differential equations?
We will solve, prove, graph and look at applications of differential equations.
A solution to a first order differential equation is an equation that does not contain a derivative.

4. Proofs of Differential Equations
Main mathematical operation - derivative

5. Ex 1: Verify that y = 1 – e-2x is a solution to the differential equation
First step – Take the derivative.
Second step - substitute into the diff equ

6. Simplify the expression and show that the 2 sides are equal.

7. Ex 2: Show that y = e7x is a solution to the differential equation

8. Ex 3: Show the y = 3e-4x-xe-4x is a solution toor

9. AP only: Example 4
For what values of ‘r’ does the function y = ert satisfy the differential equation y” + y’ – 6y = 0?

10. AP only: Example 5: Show that every member of the family of
functions is a solution of the
differential equation y’ = xy.

11. Solving Separable Differential Equations
Main mathematical operation – integration
A separable differential equation is any equation that can be written in the form P(x) dx = Q(y) dy
All ‘x’’s separated from the ‘y’s

12. Ex 1: Find the solution to the differential equation

13. Ex 2: Solve the differential equation y = f(x) given below

14. Ex 3:Find the solution to the differential equation with the initial condition y(1) = -4.

15. Ex 4:Find the solution to the differential equation with the initial condition y(0) = -4.