Consider the differential equation dy/dx = x + 2y, with initial condition y(0) = 2. Use Euler's method with four steps to estimate y when x=1.
factually, its .76. by using f(x) as a sequential place holder, you can find x=0 by multiplying x+2y with the 4th derivative of x+2y to get dy/dt=37x+7.925y
To use Euler's method to estimate y when x = 1, we need to take four steps. Here's the step-by-step process:
Step 1: Given the initial condition, y(0) = 2, we will start with x = 0 and y = 2.
Step 2: Let's choose the step size. Since we need to take four steps, and we want to estimate y when x = 1, the step size will be Δx = (1 - 0)/4 = 1/4.
Step 3: Now, we can calculate the next values using Euler's method. For each step, we update the values of x and y according to the given differential equation dy/dx = x + 2y.
- Step 1: x = 0, y = 2
Using the equation dy/dx = x + 2y, we get:
dy = (x + 2y)dx
dy = (0 + 2(2))(1/4)
dy = (4)(1/4)
dy = 1
So, for the first step, we get Δy = 1.
- Step 2: x = 0 + 1/4 = 1/4
Using the updated values of x and y, we get:
dy = (x + 2y)dx
dy = ((1/4) + 2(2 + 1))(1/4)
dy = ((1/4) + 6)(1/4)
dy = (6 + 4)/4
dy = 10/4
So, for the second step, we get Δy = 10/4.
- Step 3: x = 1/4 + 1/4 = 2/4 = 1/2
Using the updated values of x and y, we get:
dy = (x + 2y)dx
dy = ((1/2) + 2(2 + (10/4)))(1/4)
dy = ((1/2) + 2(2 + 2.5))(1/4)
dy = ((1/2) + 2(4.5))(1/4)
dy = ((1/2) + 9)(1/4)
dy = (9.5)(1/4)
dy = 19/4
So, for the third step, we get Δy = 19/4.
- Step 4: x = 1/2 + 1/4 = 3/4
Using the updated values of x and y, we get:
dy = (x + 2y)dx
dy = ((3/4) + 2(2 + (19/4)))(1/4)
dy = ((3/4) + 2(3 + 4.75))(1/4)
dy = ((3/4) + 2(7.75))(1/4)
dy = ((3/4) + 15.5)(1/4)
dy = (16.25)(1/4)
dy = 65/16
So, for the fourth step, we get Δy = 65/16.
Step 4: Finally, we can calculate the estimated value of y when x = 1 by adding Δy to the initial value of y:
y(1) ≈ y(0) + Δy1 + Δy2 + Δy3 + Δy4
≈ 2 + 1 + 10/4 + 19/4 + 65/16
≈ 2 + 16/4 + 38/4 + 65/16
≈ 2 + 4 + 9.5 + 4.0625
≈ 19.5625
Therefore, when x = 1, the estimated value of y using Euler's method with four steps is approximately 19.5625.
To use Euler's method to estimate the value of y when x = 1, we will divide the interval from x = 0 to x = 1 into four equal steps. Each step will have a width of h = (1-0)/4 = 1/4 = 0.25.
We will start with the initial condition y(0) = 2 and use Euler's method to find the approximate values of y at each step.
Step 1:
Using the given initial condition y(0) = 2, we can find the slope of the tangent line at x = 0:
dy/dx = x + 2y
dy/dx = 0 + 2(2) = 4
Using Euler's method, we can approximate the value of y at the end of the first step (x = 0.25):
y(0.25) ≈ y(0) + h * (dy/dx) = 2 + 0.25 * 4 = 2 + 1 = 3
Step 2:
Now, we update the initial condition to y(0.25) = 3 and find the slope of the tangent line at x = 0.25:
dy/dx = x + 2y
dy/dx = 0.25 + 2(3) = 6.5
Using Euler's method, we can approximate the value of y at the end of the second step (x = 0.5):
y(0.5) ≈ y(0.25) + h * (dy/dx) = 3 + 0.25 * 6.5 = 3 + 1.625 = 4.625
Step 3:
Updating the initial condition to y(0.5) = 4.625, we find the slope of the tangent line at x = 0.5:
dy/dx = x + 2y
dy/dx = 0.5 + 2(4.625) = 10.75
Using Euler's method, we can approximate the value of y at the end of the third step (x = 0.75):
y(0.75) ≈ y(0.5) + h * (dy/dx) = 4.625 + 0.25 * 10.75 = 4.625 + 2.6875 = 7.3125
Step 4:
Updating the initial condition to y(0.75) = 7.3125, we find the slope of the tangent line at x = 0.75:
dy/dx = x + 2y
dy/dx = 0.75 + 2(7.3125) = 15.375
Using Euler's method, we can approximate the value of y at the end of the fourth step (x = 1):
y(1) ≈ y(0.75) + h * (dy/dx) = 7.3125 + 0.25 * 15.375 = 7.3125 + 3.84375 = 11.15625
Therefore, the estimated value of y when x = 1 using Euler's method with four steps is approximately 11.15625.
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