Math 231: Differential Equations

1. Math 231: Differential Equations Set 3: Working with Vector Fields Notes abridged from the Power Point Notes of Dr. Richard Rubin

2. Skinner's Constant: That quantity which, when multiplied by, divided by, added to, or subtracted from the answer you get, gives the answer you should have gotten.

3. Direction Fields The text on p. 65 describes a situation that perhaps we all might want to share: Drifting along a lake in boat, probably relaxing and letting the current carry us along! Of course, the text suggests you are reading a book about direction fields!

4. Direction Fields Definition A direction field is the set of tangent lines (slopes) of a given differential equation drawn in the xy-plane for selected points (xi,yi) of the differential equation.

5. Direction Fields Y axis points N; X axis points E. In this coordinate system, our position is given parametrically by: x(t) = 10t and y(t) = 10t Can you write y as a function only of x?

6. Direction Fields In this coordinate system, our position is given parametrically by: x(t) = 10t and y(t) = 10t What function tells us the direction we travel at any instant in time? What is the derivative of y with respect to x? Chain Rule:

7. Direction Fields In this coordinate system, our position is given parametrically by: x(t) = 10t and y(t) = 10t If we draw a graph showing the direction of travel by a small arrow, we would get a picture like the one on pg 67. Since it shows direction of travel at each point in space, it is called a direction field.

8. Direction Fields x(t) = 10t and y(t) = 10t The graph on pg 67 is the direction field for our drifting boat. Suppose we start drifting from the origin of coordinate, what will be our approximate coordinates after 10 seconds? after 50 seconds? What equation describes our trajectory (y(x))?

9. Direction Fields x(t) = 10t and y(t) = 10t Suppose we change the scenario. Instead of drifting, we accelerate at 2 ft/sec2 while steering the boat due east starting from (0,0). The current is still carrying us north 10 ft and 10 ft east each second. y(t) = 10t as before. How do we determine x(t)?

10. Direction Fields We accelerate at 2 ft/sec2 while steering the boat due east starting from (0,0). The current is still carrying us north 10 ft and 10 ft east each second. y(t) = 10t as before. How do we determine x(t)? How do we find C?

11. Direction Fields We accelerate at 2 ft/sec2 while steering the boat due east starting from (0,0). The current is still carrying us north 10 ft and 10 ft east each second. y(t) = 10t as before. How do we determine x(t)? at t = 0, dx/dt = 10 before we accelerate!

12. Direction Fields We have We can find x(t) with the initial condition that x(0)=0:

13. Direction Fields The direction of travel is again given by dy/dx:

14. Direction Fields The direction of travel is again given by dy/dx: We can solve for t in terms of x:

15. Direction Fields The direction of travel is again given by dy/dx: Now:

16. Direction Fields The direction of travel is again given by dy/dx: Now:

17. Direction Fields Suppose the solution passes thru a point (a,b) in the xy plane. Then, we have the condition that y(a) = b. We know the slope at (a,b) is given by dy/dx. At (a,b), .

18. Direction Fields f(a,b) is the slope of the tangent to the solution at the point (a,b). We could choose (a,b) in such a way that we have a grid of equally spaced points in the xy plane. At each of these points, we plot slope of the solution by drawing an arrow with slope = f(a,b).