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

2. 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

3. 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

4. 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)

5. 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

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

7. 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.

8. Slope Fields
Ex. Sketch a slope field for

9. Slope Fields
Ex. Sketch a slope field for

10. 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)

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

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

13. 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