Let y be the solution to the differential equation
that passes through the point (0,0). Use Euler's Method to estimate y(1) with step size Δ x = 0.5.
We want to take steps of size 0.5 to get from x = 0 to x = 1, so there will be two steps.
This means we'll do two tangent line approximations. First we'll draw a tangent line to y at 0 and use that to estimate y(0.5).
Then we'll draw a line at x = 0.5 and use that to estimate y(1).
First Tangent Line Approximation: We're going to use a tangent line to y at x = 0 to estimate y(.5), so Δ x = .5. Since y(0) = 0 we have
yold = 0.
The slope of the tangent line at (0,0) is
Our tangent line is horizontal.
y(.5) ≅ 0.
Second Tangent Line Approximation: The idea is to use a tangent line to y at x = 0.5 to estimate y(1). However, we only have an estimate of y(0.5). We don't know its real value. This means the line we draw won't really be tangent to y at x = 0.5, but it will be pretty close.
We have the point (0.5, 0) to start from, so we let yold = 0. We still have Δ x = 0.5. To get the slope of the line we use the differential equation: