1. Differential Equations and Slope Fields By: Leslie Cade 1 st period

2. Differential Equations A differential equation is an equation which involves a function and its derivative There are two types of differential equations: – General solution: is when you solve in terms of y and there is a constant “C” in the problem – Particular solution: is when you solve for the constant “C” and then you plug the “C” into the y equals equation

3. Example of a General Solution Equation Given: Step 1: Separation of variables- make sure you have the same variables together on different sides of the equation. Step 2: Integrate both sides of the equation

4. Step 3: Notice how above there is a constant “C” added to both sides, but you can combine those constants on one side of the equation and end up with: Step 4: Solve for y

5. Example of Particular Solution Given: f(0)= 3 Step 1: Separate the variables like you would with a general solution equation. Step 2: Integrate both sides of the equation

6. Step 3: Apply the exponential function to both sides of the equation Step 4: Solve for y

7. Step 5: Plug in f(0)=3 and solve for C Step 6: Plug “C” into the y equals equation you found in step 4

8. Try Me! 1. 2.f(0)=2 3. f(0)=7 4.

9. Try Me Answer #1 1.Step 1: Separate the variables on each side of the equation Step 2: Integrate both sides of the equation Step 3: Apply the exponential function to both sides of the equation to get y by itself Step 4: Solve for y to find the general equation

10. Try Me Answer #2 2. f(0)=2Step 1: Separation of variables Step 2: Integrate both sides of the equation Step 3: Apply the exponential function to both sides of the equation Step 4: Plug in values given and solve for the constant “C”. Step 5: Plug in the constant you found in step 4 into your y equals equation to find the particular equation

11. Try Me Answer #3 3. f(0)=7Step 1: Separation of variables Step 2: Integrate both sides of the equation Step 3: Apply the exponential function to both sides Step 4: Solve for constant “C” given values f(0)=7 Step 5: Plug constant into y equals equation to find the particular equation

12. Try Me Answer #4 Step 1: Separate variables Step 2: Integrate both sides Step 3: Apply exponential function to both sides Step 4: Solve in terms of y to find the general equation 4.

13. Slope Fields Slope fields are a plot of short line segments with slopes f(x,y) and points (x,y) lie on the rectangular grid plane Slope fields are sometimes referred to as direction fields or vector fields The line segments show the trend of how slope changes at each point no slope (0) undefined

14. FRQ 2008 AB 5 Consider the differential equation where a. On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.

15. b. Find the particular solution y= f(x) to the differential equation with initial condition f(2)=0. c. For the particular solution y=f(x) described in part b, find

16. Try Me! Consider the differential equation a. On the axes provided, sketch a slope field for the given differential equation at the nine points indicated. Slope = -2 Slope = 2 Slope = 4

17. FRQ 2004 (FORM B) AB 5 Consider the differential equation a. On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.

18. b. While the slope field in part a is drawn at only twelve points, it is defined at every point in the xy-plane. Describe all points in the xy-plane for which the slopes are negative. and c. Find the particular solution y=f(x) to the given differential equation with the initial condition f(0)=0.

19. C.

20. FRQ 2004 AB 6 Consider the differential equation a. On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.

21. b. While the slope field in part a is drawn at only twelve points, it is defined at every point in the xy-plane. Describe all points in the xy-plane for which the slopes are positive. and c. Find the particular solution y=f(x) to the given differential equation with the initial condition f(0)=3.

23. Try Me! 1. f(0)=2 2. 3. 4. f(1)=-1

24. Try Me Answers 1.2.

25. 3. 4. f(1)=-1

26. Slope Field Example xyy’ = x + y -2 0 10 000 1 0 101 112

27. FRQ 2006 AB 5 Consider the differential equation where a. On the axes provided, sketch a slope field for the given differential equation at the eight points indicated.

28. b. Find the particular solution y=f(x) to the differential equation with the initial condition f(-1)=1 and state its domain. The domain is x<0

29. Review! xyy’=4x/y 4 0Und. 1-4 00 00Und. 010 1-4 10Und. 114

30. Review Continued… Step 1: Separation of variables Step 2: Integrate both sides of the equation Step 3: Solve in terms of y to find the general solution

31. Review Continued… f(1)=0Step 1: Separate the variables Step 2: Integrate both sides Step 3: Plug in f(1)=0 to find the constant “C” Step 4: Plug the constant you just found into the y equals equation

32. Bibliography http://www.math.buffalo.edu/~apeleg/mth30 6g_slope_field_1.gif ©Leslie Cade 2011