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Euler's Method for AP Calculus

By — McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus

Euler's Method provides a means of estimating the numerical solution of differential equations by a series of successive linear approximations. Represent the differential equation by y ' = f '(x, y) and the initial condition y0 = f(x0), and choose a small value, Δx, as the increment between estimates. Begin with the initial value y0, and evaluate y1 = y0 + Δx ·f '(x0, y0). Continue with y2 = y1 + Δx · f '(x1, y1), and in general, yn = yn-1 + Δx · f '(xn–1, yn–1).

Example 1

Given the initial value problem with y(0) = 1, approximate y(1), using five steps.

Step 1: The interval (0, 1) divided into five steps gives us Δt = 0.2.

Step 2: Create a table showing the iterations.

Example 2

Use Euler's method with a step size of Δx = 0.1 to compute y(1) if y(x) is the solution of the differential equation with initial condition y(0)=3

Step 1: For case of the evaluation, transform

Step 2: Create a table showing the iterations. A simple problem, stored in your calculator and modified with the new differential equation and initial condition, will allow you to generate the table quickly.

Example 3

Use Euler's Method to approximate P (4), givenwith initial condition P(0) = 4. Use an increment of Δt = 0.5.

Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus

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