Euler's Method
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:
Example
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 thex=-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, soΔx = (2 - -1)/3 = 1
Most tricky part is how to get
Δy
. We knowdy/dx ≃ Δy/Δx
, soΔy ≃ 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
andy = 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...
Last updated