Use Euler's method with step size 0.2 to estimate y(1), where y(x) is the solution of the initial-value problem.
y' = 5x + y^2, y(0)=-1.
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To use Euler's method with a step size of 0.2 to estimate y(1), we need to iterate through the steps and update the approximation of y at each step.
Given the initial value problem:
y' = 5x + y^2, y(0) = -1
Let's start by subdividing the interval [0, 1] into smaller steps of size 0.2. This gives us the following steps:
Step 1: x = 0, y = -1
Step 2: x = 0.2, y = ?
Step 3: x = 0.4, y = ?
Step 4: x = 0.6, y = ?
Step 5: x = 0.8, y = ?
Step 6: x = 1.0, y = ?
Using Euler's method, the formula to update the approximation at each step is:
y(i+1) ≈ y(i) + h * f(x(i), y(i))
where h is the step size (0.2 in this case), f(x, y) is the function given by the differential equation (5x + y^2), and (x(i), y(i)) represents the current values of x and y at step i.
Using this formula, we can now calculate the approximation of y at each step.
Step 1: x = 0, y = -1
Here, there's no need to calculate anything as we already know y = -1.
Step 2: x = 0.2, y = ?
Using the Euler's method formula:
y(1) ≈ y(0) + 0.2 * (5(0) + (-1)^2)
≈ -1 + 0.2 * (0 + 1)
≈ -1 + 0.2
≈ -0.8
Step 3: x = 0.4, y = ?
Using the Euler's method formula:
y(2) ≈ y(1) + 0.2 * (5(0.2) + (-0.8)^2)
≈ -0.8 + 0.2 * (1 + 0.64)
≈ -0.8 + 0.2 * 1.64
≈ -0.8 + 0.328
≈ -0.472
Step 4: x = 0.6, y = ?
Using the Euler's method formula:
y(3) ≈ y(2) + 0.2 * (5(0.4) + (-0.472)^2)
≈ -0.472 + 0.2 * (2 + 0.222784)
≈ -0.472 + 0.2 * 2.222784
≈ -0.472 + 0.4445568
≈ -0.0274432
Step 5: x = 0.8, y = ?
Using the Euler's method formula:
y(4) ≈ y(3) + 0.2 * (5(0.6) + (-0.0274432)^2)
≈ -0.0274432 + 0.2 * (3 + 0.000751573049)
≈ -0.0274432 + 0.2 * 3.000751573049
≈ -0.0274432 + 0.60015031461
≈ 0.57270711461
Step 6: x = 1.0, y = ?
Using the Euler's method formula:
y(5) ≈ y(4) + 0.2 * (5(0.8) + (0.57270711461)^2)
≈ 0.57270711461 + 0.2 * (4 + 0.32752099337)
≈ 0.57270711461 + 0.2 * 4.32752099337
≈ 0.57270711461 + 0.865504198674
≈ 1.43821131328
Therefore, the estimate of y(1) using Euler's method with a step size of 0.2 is approximately 1.43821131328.
To use Euler's method to estimate y(1) for the given initial-value problem, we need to follow these steps:
Step 1: Determine the number of subintervals.
In this case, the step size is given as 0.2, and we need to find y(1). So, there are 5 subintervals, with h = (1-0)/5 = 0.2.
Step 2: Define the initial values.
Given that y(0) = -1, we start with (x_0, y_0) = (0, -1).
Step 3: Define the formula for Euler's method.
Euler's method states that y_{i+1} = y_i + h * f(x_i, y_i), where
- y_{i+1} is the approximation of y at the next subinterval,
- y_i is the value of y at the current subinterval,
- h is the step size,
- f(x_i, y_i) is the value of the derivative function at the current subinterval.
Step 4: Iterate for each subinterval.
Using the given differential equation y' = 5x + y^2, we can calculate the value of f(x_i, y_i) at each subinterval.
We can now proceed with the calculations:
For i = 0:
- x_0 = 0
- y_0 = -1
- f(x_0, y_0) = 5(0) + (-1)^2 = -1
Using the formula for Euler's method:
y_1 = y_0 + h * f(x_0, y_0)
y_1 = -1 + 0.2 * (-1) = -1 - 0.2 = -1.2
For i = 1:
- x_1 = x_0 + h = 0.2
- y_1 = -1.2
- f(x_1, y_1) = 5(0.2) + (-1.2)^2 ≈ 3.44
Using the formula for Euler's method:
y_2 = y_1 + h * f(x_1, y_1)
y_2 = -1.2 + 0.2 * 3.44 = -1.2 + 0.688 ≈ -0.512
Repeat these steps until we reach the desired subinterval (i = 5).
For i = 2, y_3 ≈ -0.118
For i = 3, y_4 ≈ 0.579
For i = 4, y_5 ≈ 1.791
For i = 5, y_6 ≈ 3.794
Therefore, using Euler's method with a step size of 0.2, the estimated value of y(1) for the given initial-value problem is approximately 3.794.