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.

Refer to Wiki: Euler method

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 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...