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Section 9.1 – Introduction to Differential Equations What is a “Differential Equation”? It is an equation with a “differential” in it. Duh. Some examples: y ' 3x y y ' x y 6 2 y " 3 y ' 4 y 3 2 d y dy 4x y 2 dx dx 2 2 dy 1 2 dx 1 x Where are they used? In Physics, you learn about falling motion and the acceleration due to gravity, a constant acceleration downwards. Another factor is wind resistance. But this is not constant; it depends on the velocity of the object. This is an example where the change in velocity (acceleration) depends on the velocity at which it is traveling. There are lots of other uses in Physics, Engineering, City Planning and other occupations. Visualization of heat transfer in a pump casing, by solving the heat equation. Determine whether y 3e x 2e 2x is a solution to the differential equation y " 3y ' 2y 0. y ' 3e x 4e2x y " 3e x 8e2x x 2x x 2x x 2x 3e 8e 3 3e 4e 2 3e 2e 0 3e x 8e2x 9e x 12e2x 6e x 4e2x 0 18e x 24e2x 0 NO Determine whether y 4e x e2x is a solution to the differential equation y " 3y ' 2y 0. y ' 4ex 2e2x y " 4ex 4e2x 4e x 4e2x 3 4ex 2e2x 2 4ex e2x 0 4ex 4e2x 12ex 6e2x 8ex 2e2x 0 0 0 YES Notice both of those problems used the exponential function. Why do you think this is? When we solve things, we are going to need derivatives to cancel out original functions. This happens often with the exponential function because its derivative is itself. There are entire college classes that deal with Differential Equations, but we will only touch on the basics.