Reading Between the Lines! Slope Fields and Euler’s Method

Slope Fields A first order differential equation can be written as a condition in the form in which the slope of the function can be calculated as a function of x and y

What function or family of functions produces a slope field like this? This sometimes enables us to determine the correct form of y(x). For example: What function or family of functions produces a slope field like this?

What function or family of functions produces a slope field like this?

Your turn… Sketch slope fields for the following and try to guess the mathematical function or relation that these represent: a b c

Euler’s Method… Slope fields also suggest an algebraic approximation scheme that we could use to numerically solve a differential equation

Example… Compare an Euler’s solution of with an exact solution. Use h = 0.1 and plot the solution between x = 0 and x = 10. Use Excel.

Some Worked Examples… Look at 11.3 # 8,9 in Excel

Working With DEs and Separation of Variables Method is very useful for many cases in which variables can be isolated from one-another Requires that isolated (separated) terms are integrable If separation doesn’t work can always fall back on Euler’s Method Examples from 11.4; 1, 15, 27