1. Lesson 9-2 Direction (Slope) Fields and Euler’s Method

2. Solutions to Differential Equations The solution to a differential equation is a function and it may be general or particular (given an initial condition) There are 3 ways to solve a differential equation: –Analytically (Separate & Integrate) –Graphically (with Slope Fields) –Numerically (with Euler’s Method)

3. Direction or Slope Field The directional field allows us to visualize the general shape of the solution curves by indicating the direction (the slope at that point) in which the curves proceed at each point. We draw the curve, given an initial condition, so that it is parallel to the nearby line segments (slopes in the field)

4. f(0) = -2 Slope Field Example

5. Example 1 Given: dy -xy² ---- = -------- with f(-1) = 2 dx 2 Graph the slope field at the twelve indicated points

6. Example 2 Given: dy -2x ---- = -------- with f(1) = -1 dx y Graph the slope field at the twelve indicated points

7. Example 3 Given: dy ---- = x 4 (y – 2) with f(0) = 0 dx Graph the slope field at the twelve indicated points

8. Example 4 Given: dy ---- = x 2 (y – 2) with f(0) = 3 dx Graph the slope field at the twelve indicated points

9. Euler’s Method Euler’s method is an iterative process (like fractals in Geometry) in which the next value is dependent on the previous value. It is used in many different applications and is the foundation of an area of mathematics called Time Series Analysis. Some hurricane models are based on these same type of algorithms. For the first estimate: y 1 = y 0 + h F(x 0, y 0 ) (Stewart’s notation: h is ∆x and F is slope) y 1 = y 0 + slope(x 0,y 0 ) ∆x (from the notation in your notes) For the nth estimate: y n = y n-1 + h F(x n-1, y n-1 ) y n = y n-1 + slope(x n-1,y n-1 ) ∆x These iterative processes are just what spreadsheets were made for. The next slide shows example 3 on page 597 of Stewart

10. Example 3 page 597 islopexy∆x=0.1 0101 11.20.11.1 21.420.21.22 31.6620.31.362 41.92820.41.5282 52.221020.51.72102 62.5431220.61.943122 72.89743420.72.197434 83.287177620.82.487178 93.7158953820.92.815895 104.1874849213.187485 = x + y = xi + ∆x =yi-1 + ∆xslopei-1

11. Euler Example 1 Given with y(0) = 1; Use Euler’s method starting at x = 0 with a step size of 0.2 (∆x) to approximate y(1). y n = y n-1 + slope(x n-1,y n-1 ) ∆x dy x ----- = ------ dx y² islopexy∆x=0.2 0001 1 1 20.3698220.41.04 30.4835130.61.113964 40.5458090.81.210667 50.5740711.319829

12. Euler Example 1 cont Now find the particular solution to and find y(1). Compare the exact and approximate values. Remember the steps to solve simple differential equations! dy x ----- = ------ dx y² dy x ----- = ------ dx y² y² dy = x dx∫ y² dy = ∫ x dx ⅓y³ = ½x² + C y = (3/2)x² + C y(0) = 1 so C = 1 y(1) = (3/2) + 1 = 1.3572 vs 1.319829 3 3 with ∆x = 0.1 Euler’s Method = 1.339315 Important to note that the C is under the cube root!!

13. Summary & Homework Summary: –Slope field is just the slope of the function, the value of the derivative graphed at the point –Euler’s method, an iterative method, allows us to solve for the function analytically Homework: –pg 599 – 601: 3-7, 11, 12, (Euler Problem 23)