Mathematics 4330 / 5344 – # 6 Introduction to Solving Differential Equations

Euler's one step method is undoubtedly the simplest method for approximating the solution to an ordinary differential equation. It goes something like this: Given a first order initial value problem y = f (x, y), x ∈ (a, b) y(a) = y a we observe that defining x j = a + h(j − 1), for j = 1, 2, · · · , (N + 1) where h = (b − a)/N we have y(x k+1) − y(x k) h ≈ f (x k , y(x k)). Therefore we can write y(x k+1) ≈ y(x k) + h f(x k , y(x k)). With this we can define an iterative scheme for computing approximate values y k for y(x k) by y k+1 = y k + f (x k , y k), k = 1, 2, · · · , (N + 1). Suppose we have a function file de_fn.m, say, function z=de_fn(x,y) z=-x*y; and we build the m-file eul.m