1. AP Calculus Warm up
3.12.13
The acceleration of a particle is given by:
a (t )
32
Find a) The velocity function
b.) The position function
2. Differential Equations
âą Equations that have derivatives.
âą The order , is the highest derivative overall.
First order â differential equation:
y
xy
3
Second order â differential equation
y
y
0
âą You can solve a differential equation by integrating
both sides (Integration and Differentiation are inverse
operations)
âą The solution to a differential equation is also an
equation.
3. How do we verify if an equation
is a valid solution to a differential equation?
Substitute and see if it satisfies the equation.
y
Is :
y
y
sin x
0
a solution?
Is :
y
4e
x
a solution?
4. Finding a particular solution
âą For the differential equation:
a. Verify that y
3
xy
3y
0
is a solution.
b. Find the particular solution determined by the
initial condition y = 2 when x = -3
Cx
5. Practice
âą For the differential equation:
a. Verify that y
2x
y
y
2y
is a solution.
b. Find the particular solution determined by the
initial condition y = 5 when x = 0
Ce
0
6. Slope fields (direction field)
âą Solving a differential equation can sometimes
be difficult or impossible.
âą Slope fields give us a graphical approach to
solving.
âą To do it, the differential equation needs to be
solved for the first derivative (For example)
y
x 1
dy
or
sin x
2
dx
âą Since the first derivative gives us the slope, we
can get the slope of the solution at any point.
7. How to create a slope field (direction field)
âą Find the slope at each Given point by plugging
into the derivative.
âą Draw a short line segment representing the
slope at those points.
âą The slope field shows the general shape of all
the solutions.
8. Example
â Example: Sketch a slope field for the differential
equation y x y
Use the points ( -1,1), (0,1) , and (1,1)
9. Example 2
âą Sketch the slope field for the differential equation:
y
2 x y through the following points:
(-2,2)
(-2,1)
(-1,-1)
(-1,1)
(0,-1)
(0,1)
(1,-1)
(1,1)
(2,-1)
(2,1)