Ms. Battaglia AP Calculus

Estimate y(4) with a step size h=1, where y(x) is the solution to the initial value problem: y’ – y = 0 ; y(0) = 1

Euler’s method is a numerical approach to approximating the particular solution of the differential equation y ’ = F(x,y) that passes through the point (x 0,y 0 ). Starting point: the graph of the solution passes through the point (x 0,y 0 ) and has the slope F(x 0,y 0 ). Next, proceed in the direction indicated b the slope. Using a small step h, move along the tangent line until you arive at the point (x 1,y 1 ) where x 1 = x 0 + h and y 1 = y 0 + hF(x 0,y 0 ) If you think of (x 1,y 1 ) as a new starting point, you can repeat the process to obtain a second point (x 2,y 2 )

x 1 = x 0 + hy 1 = y 0 + hF(x 0,y 0 ) x 2 = x 1 + hy 2 = y 1 + hF(x 1,y 1 ).. x n = x n-1 + h y n = y n-1 + hF(x n-1,y n-1 )

Use Euler’s Method to approximate the particular solution of the differential equation y ‘ = x – y passing through the point (0,1). Use a step of h=0.1 and n=10.

Use Euler’s Method to approximate the particular solution of the differential equation y ‘ = x + y passing through the point (0,2). Use a step of h=0.1 and n=10.

Use Euler’s Method to approximate the particular solution of the differential equation y ‘ = 3x – 2y passing through the point (0,3). Use a step of h=0.05 and n=10.

 AB: Page 413 #73, 74, 77, 78 (use n = 5 for all 4 problems), 79,  BC: Page 413 #73-78,89-92, and Worksheet(matching)