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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  4e2x  y "  3e x  8e2x
x
2x
x
2x
x
2x

3e

8e

3
3e

4e

2

3e

2e

 
 
0
3e x  8e2x  9e x  12e2x  6e x  4e2x  0
18e x  24e2x  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.