Euler's method is a numerical approach for approximating solutions to differential equations. It works by taking an initial condition and using the tangent line at that point to take a small step to a new point. This process is repeated, using the new point as the initial condition. The smaller the step size, the more accurate the approximation will be. An example walks through applying Euler's method to the differential equation y' = x + y with an initial condition of y(0) = 2 using 10 steps of size 0.1.

1. Euler’sMethodCathie DominskiZoe HsiehLinda HongStella JohPeriod 3

2. What is Euler’s [OIL-er’s] Method?Numerical approach to approximating the particular solution of the differential equationy’=F(x,y)TRANSLATION: Essentially, you take an initial condition (a point), and use that condition as a starting point to take a small step h along the tangent line to get to a new point. Rinse and repeat using the new point as a starting point.*Note: The smaller the steps used, the more accurate your graph will be.

3. Explanation of Euler’s Methody’= F(x,y) w/ initial condition (x0,y0)x1= x0+h (h is your step)y1=y0+hF(x0,y0)Plug your initial condition into this formula to find (x1,y1) and RINSE AND REPEATthis is the slope of the tangent line at your initial point

4. F(x,y)How to write a first-order differential equation in the formExample:is rewritten asso that

5. yTrue Valueyi+1, Predicted value ΦyihStep sizexxixi+1Euler’s Method (standard)

6. Example 1: page 411 #69Use Euler’s Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use n steps of size h. y’=x+y y(0)=2 n=10 h=0.1

7. Example 1: page 411 #69Y(0)=2N=10H=0.1 y1=y0+hF(x0,y0) y1=2+(0.1)(0+2) y1=2.2 y2=y1+hF(x1,y1) y2=2.2+(0.1)(0.1+2.2) y2=2.43

8. Example 1: page 411 #69

9. Real Life Application…In 2000, the national debt of the United States was 5.6 trillion dollars and in 2001, the debt rose to 5.8 trillion dollars. Using Euler’s method, one could have estimated the national debt for the following years.t= years where t=0 is 2000, w=debt in billions of dollarsIn year 2000 $5.6 trillion  in 2001 $5.8 trillion is a 2% increase.w’=.02wUsing step size h=1 for the range (0,10) the first step would look something like…T0=0 W0=5628.7W1=W0+F(t0,w0)(h)5628.7+(.02 x 5628.7)(1)w(1)=$5741.3 in billions.So 2001, the estimated value of the national debt is about 5.7 trillion dollars.Still pretty close to the actual value…

10. If you keep going…You eventually get the estimated value for the annual debt of the United States up to year 10, 2010.However…, as you solve for more values of w, it gets more inaccurate because Euler’s method is just an estimation.By looking at the actual values of each year’s debt we can see the inaccuracy of the Euler’s method for this step size.

12. FIN!“No Tragic Mistakes!”