LEGACY Numerical Methods

A-Level Further Pure

First order differential equations

Some differential equations can not be solved analytically; however, they can be accurately approximated with Euler’s Method:

The way this works is that you are given initial conditions . So you can compute , then you can approximate the coordinates some away from by using a straight line with gradient . To get a more accurate approximation, you can calculate the next point with smaller steps . As a result, you have to do more iterations to get a point some distance from

You could also use the Midpoint Formula:

Second order differential equation

You can extend Euler’s method to second order differential equations with:

Simpson’s Rule

You can approximate with the formula:

where is the number of strips used to approximate the area and is the width of each strip. This formula is derived by using quadratic curves and using the result that area of quadratic curve passing through , , is .

Other forms:

This formula approximates the area under the curve by using quadratic curves to fit the function, providing a more accurate estimate than simpler methods like the trapezoidal rule.

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