Let y = f(x) be the solution to the differential equation dy/dx = y-x The point (5,1) is on the graph of the solution to this differential equation. What is the approximation of f(6) if Euler’s Method is used given ∆x = 0.5
To approximate the value of f(6) using Euler's Method, we will use the following steps:
Step 1: Determine the differential equation dy/dx = y - x
Step 2: Set up the initial condition y(5) = 1, which means that the point (5,1) is on the graph of the solution.
Step 3: Choose a step size, ∆x = 0.5, which represents the change in x as we move forward.
Step 4: Use Euler's Method to find the approximations for f(6) using the following iteration:
- Start with the initial condition: (x0, y0) = (5, 1)
- Compute the slope: m = y - x = f(x) - x
- Calculate the change in y: ∆y = m * ∆x = (f(x) - x) * ∆x
- Calculate the next y-value: y1 = y0 + ∆y
Step 5: Repeat the iteration for each step until we reach the desired value of x, which is x = 6 in this case.
Let's go through the steps to find the approximation of f(6) in more detail:
Step 1: The given differential equation is dy/dx = y - x.
Step 2: The initial condition is y(5) = 1, which means the starting point is (5, 1).
Step 3: The chosen step size is ∆x = 0.5.
Step 4: We will use Euler's Method to calculate the successive approximations for f(6) as follows:
Iteration 1:
- Start with the initial condition: (x0, y0) = (5, 1)
- Compute the slope: m = y - x = f(x) - x = y0 - x0 = 1 - 5 = -4
- Calculate the change in y: ∆y = m * ∆x = -4 * 0.5 = -2
- Calculate the next y-value: y1 = y0 + ∆y = 1 - 2 = -1
Iteration 2:
- Start with the previous point: (x0, y0) = (5, -1)
- Compute the slope: m = y - x = f(x) - x
- In this case, we have to calculate f(x) at x = 5 because we don't have an explicit equation for f(x).
- Using the given differential equation, dy/dx = y - x, we can substitute y = f(x) into the equation: df(x)/dx = f(x) - x.
- Therefore, at x = 5, dy/dx = f(5) - 5.
- Calculate the change in y: ∆y = m * ∆x = (f(5) - 5) * 0.5 = (y0 - x0) * ∆x = (-1 - 5) * 0.5 = -6
- Calculate the next y-value: y1 = y0 + ∆y = -1 - 6 = -7
Step 5: We continue this process, repeating the iterations until we achieve the desired x-value, which is x = 6.
In summary, using Euler's Method with a step size of ∆x = 0.5, we find that the approximation of f(6) is -7.
To approximate the value of f(6) using Euler's method, we need to follow these steps:
1. Start with the given initial condition (5, 1) on the graph of the solution.
2. Calculate the slope of the tangent line at this point. In this case, the slope is given by dy/dx = y - x, so substituting x = 5 and y = 1, we get dy/dx = 1 - 5 = -4.
3. Multiply the slope by the interval ∆x = 0.5 to find the change in y (∆y). So, ∆y = -4 * 0.5 = -2.
4. Add the change in y (∆y) to the initial y-value (1) to find the next y-value. In this case, the new y-value would be 1 + (-2) = -1.
5. Repeat steps 2-4 at the new point (x = 5 + ∆x, y = -1) to find the next point on the graph.
6. Continue this process iteratively until you reach the desired x-value of 6.
Let's perform the iterations:
- Step 1: Initial Condition - f(5) = 1
- Step 2: Slope at (5, 1) - dy/dx = 1 - 5 = -4
- Step 3: ∆y = -4 * 0.5 = -2
- Step 4: f(6) = f(5) + ∆y = 1 + (-2) = -1
Therefore, the approximation for f(6) using Euler's method with ∆x = 0.5 is -1.
I already did this for you.
You know that the slope of the secant is m=∆y/∆x
So, ∆y = m*∆x
But for small ∆x, ∆y/∆x ≈ dy/dx.
So, ∆y ≈ y' * ∆x
That means f(5.5) = f(5) + y' * 0.5
f(6) = f(5.5) + y' * 0.5
You know y' = y-x, so now just plug in your numbers and iterate the two steps