"Given dy/dx=x+2y and y=0 when x=1, use Euler's method with increments of deltax=.2 to approximate when x=1.6. Show all work"
I would also like to see the answer to this one!
so, just apply the method three times.
y(x+h) = y(x) + y'(x)*h
y(1.2) = y(1) + (1+2*0)(0.2) = 0.6
y(1.4) = y(1.2) + (1.2+2*0.6)(0.2) = 1.08
and so on
Can you please complete the question? I am still slightly confused.
I gave you the formula and two examples.
You need to read up on Euler's method. It just relies on the fact that given the tangent line at (x,y), ∆y/∆x ≈ dy/dx
To use Euler's method to approximate the value of y at a specific point, we start with an initial value (x0, y0) and use the equation dy/dx = f(x, y) to calculate the value of y at the next point. Euler's method uses small increments (deltax) to approximate the change in y, based on the slope of the function at each point.
Let's break down the steps to approximate y when x = 1.6 using Euler's method with deltax = 0.2.
Given information:
dy/dx = x + 2y
y = 0 when x = 1
deltax = 0.2
Step 1: Find the initial value (x0, y0)
In this case, we are given that y = 0 when x = 1. So our initial value is (1, 0).
Step 2: Calculate the slope at the initial point, f(x0, y0)
Using the given equation dy/dx = x + 2y, substitute x0 = 1 and y0 = 0:
f(1, 0) = 1 + 2(0) = 1
Step 3: Calculate the approximate value of y at the next point
Using Euler's method, we can now find the approximate value of y at the next point (x1, y1):
y1 ≈ y0 + deltax * f(x0, y0)
≈ 0 + 0.2 * 1
Step 4: Update the values of x and y
In this case, our initial point is (x0, y0) = (1, 0), and the updated point is (x1, y1).
x1 = x0 + deltax = 1 + 0.2 = 1.2
y1 = 0.2
Step 5: Repeat steps 2-4 until you reach the desired x-value
Continue the process using the updated x1 and y1 as the new values of x0 and y0:
For x = 1.4:
f(1.2, 0.2) = 1.2 + 2(0.2) = 1.6
y2 ≈ 0.2 + 0.2 * 1.6 = 0.52
(x2, y2) = (1.4, 0.52)
For x = 1.6:
f(1.4, 0.52) = 1.4 + 2(0.52) = 2.44
y3 ≈ 0.52 + 0.2 * 2.44 = 0.988
(x3, y3) = (1.6, 0.988)
Therefore, when x = 1.6, the approximate value of y using Euler's method with deltax = 0.2 is 0.988.