Differential Equations

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Presentation transcript:

Differential Equations By Johnny Grooms and Ryan Barr AP Calculus BC Mrs. Miller, 2nd Period

Basic Differential Equations When solving, the answer is an equation Solve The answer is a general solution with a constant Answer: If given a certain condition such as (1,2) then find a particular solution 2 = 1+c and therefore c=1 The answer would be

Separable Differentiable Equations Rewrite so that y and dy are on the same side of the equation opposite x and dx. can be rewritten to be Then take the Integral of each side of the equation Remember to write + c on the x-side of the equation in order to account for the constant that is formed when taking an integral

Separable Differentiable Equations Remember that after taking the integral of each side of the equation, you might end up with the equation where there is y= , substitute A in for this and then solve the equation using conditions given. Find the general solution and also a particular solution for (4,2)

More on Differential Equations and Word Problems If y is directly proportional to x then If the rate of change of y is directly proportional to y then The general solution for this is

More on Differential Equations and Word Problems Acceleration, Velocity, Speed, and Distance Net Distance: Total Distance: a=dv/dt v=dS/dt

Slope Fields At a point on the graph, draw short lines of the relative slope of the equation at the point Used to visualize the direction of the slope of the equation May also be given a point in which to sketch a solution for on the slope field.

Slope Fields Ex. Sketch a slope field for

Slope Fields Ex. Sketch a slope field for

Euler’s Method Non-Table Form When given a derivative and a point… Make an equation for the tangent line at the point (Further steps and examples will be on the board)

Euler’s Method Table Form If given , and then using two steps… x y’ = m y 1 7 -4

Logistic Growth Equation: After solving… A is carrying capacity K is growth factor See the board for more notes on Logistic Growth

By Johnny Grooms and Ryan Barr AP Calculus BC Mrs. Miller, 2nd Period The End By Johnny Grooms and Ryan Barr AP Calculus BC Mrs. Miller, 2nd Period