AP Calculus Warm up3.12.13

The acceleration of a particle is given by:

Find a) The velocity functionb.) The position function

32)( ta

Differential Equations

• Equations that have derivatives.• The order , is the highest derivative overall.

First order – differential equation:

Second order – differential equation

• 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 yxy

0 yy

How do we verify if an equation is a valid solution to a differential equation?

0 yy

Is : a solution? xy sin

Substitute and see if it satisfies the equation.

Is : a solution? xey 4

Finding a particular solution

• For the differential equation: a. Verify that is a solution. b. Find the particular solution determined by the initial condition y = 2 when x = -3

03 yyx3Cxy

Practice

• For the differential equation: a. Verify that is a solution. b. Find the particular solution determined by the initial condition y = 5 when x = 0

02 yyyxCey 2

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)

or

• Since the first derivative gives us the slope, we can get the slope of the solution at any point.

1 xy 2sin xdx

dy

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.

Example

– Example: Sketch a slope field for the differential equation

Use the points ( -1,1), (0,1) , and (1,1)yxy

Example 2• Sketch the slope field for the differential equation:

through the following points:yxy 2

(-2,2)(-2,1)(-1,-1)(-1,1)(0,-1)(0,1)(1,-1)(1,1)(2,-1)(2,1)