1. Slope Fields Differential Equations

2. Slope Fields A slope field is a graphical picture of a derivative that projects the curve within the picture. Or a bunch of little line segments that show the slope of the curve (y = ) at different points.

3. except for the constant  Recall that indefinite integration, or antidifferentiation, is the process of reverting a function from its derivative. In other words, if we have a derivative, the antiderivative allows us to regain the function before it was differentiated – except for the constant, of course.  If we are given the derivative dy / dx = f ‘ ( x ) and we solve for y (or f ( x )), we are said to have found the general solution of a differential equation.  For example: Let Slope Fields And we can easily solve this: This is the general solution:

4. Slope Fields We can see that there are several different parabolas that we can sketch in the slope field with varying values of C  When we solve a differential equation this way, we are using an analytical method. slope fields direction fields  But we could also use a graphically method; the graphical method utilizes slope fields or direction fields.  Slope fields basically draw the slopes at various coordinates for differing values of C.  For example, the slope field for dy/dx = x is:

5. Given: and when, find the equation for. This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.

6. Initial value problems and differential equations can be illustrated with a slope field. Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but a recent AP test asked students to draw a simple one by hand.

8. We have differential equation… now what? 1. Select a coordinate (x,y) and plug it into the diff. equation. 2. Evaluate at that point (x,y) and you will get the slope at (x,y). 3. Draw a short line segment at point (x,y) on the coordinate plane with that slope. 4. Do this until you have enough points to draw a solution curve through all the points. 5. If given an initial point, then draw graph with the initial point on the graph.

9. Some helpful tips before we start  Make a chart of points (x,y) and their respective slopes to make it easier for you to graph.  Draw line segments in all 4 quadrants.  Draw enough line segments so you can predict the slopes of other points without evaluating.

10. Slope Fields  Graph of slopes of differential equations.  Draw small dash at each point (x,y).  Small dash should have slope equal to value of differential equation at (x,y)  (x,y) slope  1,1 1  1,2 2  2,1 ½  2,2 1  2,4 2  4,2 ½

11. Example 1  (x,y) slope  1,1 1  1,2 2  2,1 ½  2,2 1  2,4 2  4,2 ½  0,aundefined  a,0 0

12. Given: Let’s sketch the slope field …

13. But how?  Substitute the x and y into the differential equation for each of the points.  Plot this slope on the graph.

15. i)1 st the slopes will be positive when the differential equation is positive, i.e. ii)And x 2 is always positive (except when x=0) iii) y – 1 > 0 when y > 1 ANSWER: y > 1, but x ≠ 1 When are the slopes positive?

16. y’ = x - y Example  (x,y)slope  (0,1)-1  (3,1)2  (1,3)-2  (4,0)4  (0,3)-3  (2,2)0  (a,a)0

17. Slope Fields  Let’s examine how we create a slope field.  For example, create the slope field for the differential equation (DE): Since dy/dx gives us the slope at any point, we just need to input the coordinate: At (-2, 2), dy/dx = -2/2 = -1 At (-2, 1), dy/dx = -2/1 = -2 At (-2, 0), dy/dx = -2/0 = undefined And so on…. This gives us an outline of a hyperbola

18. Slope Fields  Let’s examine how we create a slope field.  For example, create the slope field for the differential equation (DE): Of course, we can also solve this differential equation analytically:

19. Slope Fields  For the given slope field, sketch two approximate solutions – one of which is passes through the given point: Now, let’s solve the differential equation passing through the point (4, 2) analytically: Solution:

20. C Slope Fields In order to determine a slope field from a differential equation, we should consider the following: isoclines i) If isoclines (points with the same slope) are along horizontal lines, then DE depends only on y ii) Do you know a slope at a particular point? iii) If we have the same slope along vertical lines, then DE depends only on x iv) Is the slope field sinusoidal? v) What x and y values make the slope 0, 1, or undefined? vi) dy/dx = a( x ± y ) has similar slopes along a diagonal. vii) Can you solve the separable DE? 1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____ 7. _____ 8. _____ Match the correct DE with its graph: AB C E G D F H H B F D G E A

21. How to pick out a multiple choice answer (equation) for a slope field…  Pay attention to whether you need just the x- or y- values or both  Look for places where the slope is 0  Look at the slopes along the x-axis (where y = 0)  Look for slopes along the y-axis (where x = 0)  Notice where the slopes are positive and where they are negative

22. The thought process… i)Isoclines along horizon. lines → DE depends only on y ii)Same slope along vert. line → DE depends only on x iii)Sinusoidal iv)Consider a specific point – what is the slope there. v)What makes the slope zero? What values of x and y make the slope 1? vi)Note y'(x) = x+y has same slopes along the diagonal. vii)Solve the separable DE E C A G B D I H F K J

23. Multiple Choice Example: Choose the slope field of the following graph with particular solution (0,0) to the correct differential equation. a) y’=y+x b) y’=y-x c) y’=x^2 d) y’=7-x  Graph of differential equation:

24. And the answer is…  A) is the correct answer!

25. a) f’(x)=y b) f’(x)=y+x c) f’(x)=sin(x+y) d) f’(x)=-x/y

26. And the answer is…  1. b)  2. a)  3. d)  4. c)

27. Go to: and enter the equation as:Y= For more challenging differential equations, we will use the calculator to draw the slope field. (Notice that we have to replace x with t, and y with y1.) (Leave y1 blank.) On the TI-89: Push MODE and change the Graph type to DIFF EQUATIONS. MODE Go to:Y= Press and make sure FIELDS is set to SLPFLD. I

28. Set the viewing window: Then draw the graph: WINDOW GRAPH

29. Be sure to change the Graph type back to FUNCTION when you are done graphing slope fields.

30. The TI-Nspire CAS has the capability to draw slope fields. Open a new graph page and press menu, Graph Type, Diff Eq