Section 11.3 Euler’s Method

We have seen how to draw a solution curve given a slope field and some initial value Now we are going to see how to find solutions to differential equations by calculating the slope at a point and using it to guide us to the next point At the next point we will recalculate the slope and see where our next point is We continue to do this to find our numerical solution This is called Euler’s Method

Let’s apply Euler’s Method to We will start at the point (1, 0) and find the solution for when x = 2 Using 2 steps Using 5 steps

Let’s apply Euler’s Method to We will start at the point (1, 0) and find the solution for when x = 2 Using 2 steps Using 5 steps

Let’s revisit our Yam problem with our temperature of 20° when t = 0 for the first 5 minutes dT/dt: .025(200-T) Exact Solution: 200-180e-.025t 20 4.50 20.00 1 24.5 4.39 24.44 2 28.89 4.28 28.78 3 33.17 4.17 33.01 4 37.34 4.07 37.13 5 41.41 3.96 41.15

When is Euler’s method an overestimate and when is it an underestimate? Remember how it works: It uses the rate of change locally It basically uses tangent lines for estimates