Published on March 2, 2014
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
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.
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?
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
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
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.
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 y x y Use the points ( -1,1), (0,1) , and (1,1)
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)
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