Euler's Method
Last updated
Last updated
Euler's method means an approximation by writing down every critical value in a table, and iterate many many times until it get closer to the target value.
Approximation:
Iterate table:
Solve:
It need quite a few ticks. But let's see the result first:
The table above is the Euler's Method of approximation.
As the Euler's Method, we need to figure out how to get each column value, and iterate every row.
Let's see the Initial row (R₀):
We have the Initial Condition, so for the initial row, We know the x=-1, y=3
And for iteration, we really need to know how much will the x & y change, and they change differently.
We've given that x is from -1 to 2 in 3 steps, so Δx = (2 - -1)/3 = 1
Most tricky part is how to get Δy. We know dy/dx ≃ Δy/Δx, so Δy ≃ dy/dx · Δx.
Under the initial condition, dy/dx = (-1) - (3) - 2 = -6
So for this iteration, Δy = dy/dx · Δx = -6 × 1 = -6
Now we get everything for first round (iteratioin):
We let x = -1 +(1) = 0 and y = 3 +(-6) = -3
For this round, dy/dx = x - y - 2 = 0 - (-3) -2 = 1
So in this round, Δy = dy/dx · Δx = 1 × 1 = 1
And let's get into the second round..
Third round...
