The document provides information about solving differential equations including: - Finding the velocity and position functions given an acceleration function - Identifying orders of differential equations - Verifying solutions by substituting back into the original equation - Finding particular solutions given an initial condition - Using slope fields to graphically represent solutions when an equation cannot be solved analytically

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
• Equations that have derivatives.
• The order , is the highest derivative overall.
First order – differential equation:
y
xy
3
Second order – differential equation
y
y
0
• You can solve a differential equation by integrating
both sides (Integration and Differentiation are inverse
operations)
• 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
• 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
• 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)
• Solving a differential equation can sometimes
be difficult or impossible.
• Slope fields give us a graphical approach to
solving.
• 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
• 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)
• Find the slope at each Given point by plugging
into the derivative.
• Draw a short line segment representing the
slope at those points.
• The slope field shows the general shape of all
the solutions.

8. Example
– 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
• 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)