DIFFERENTIAL EQUATIONS

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Differential Equations

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Presentation transcript:

DIFFERENTIAL EQUATIONS

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

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.

Proofs of Differential Equations Main mathematical operation - derivative

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

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

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

Ex 3: Show the y = 3e-4x-xe-4x is a solution to or

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

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

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

Ex 1: Find the solution to the differential equation

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

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

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