An initial-value problem is given by the differential equation,
f(x,y)=x(1-y^2), y(1)=0.07
Use the Euler-trapezoidal method with a step-size h = 0.1, to obtain the approximate value of y(1.1). Give your answer to 4 decimal places.
To approximate the value of y(1.1), we can use the Euler-trapezoidal method. This method involves taking small steps from the initial value of y(1) to the desired value of y(1.1) using the given differential equation.
Step 1: Calculate the number of steps needed. Since the step-size h is equal to 0.1, we need to take 11 steps to reach x = 1.1, starting from x = 1.
Step 2: Initialize the variables for the first step. Set x0 = 1, y0 = 0.07, and h = 0.1.
Step 3: Begin the iteration to approximate the value of y(1.1). For each step, follow these substeps:
3.1: Calculate f(x, y) using the given differential equation. In this case, f(x, y) = x(1 - y^2).
3.2: Calculate the Euler increment using the current values of x, y, and h. The Euler increment is given by (h * f(x, y)) / 2.
3.3: Update the values of x and y using the Euler increment. Set x = x + h and y = y + Euler increment.
3.4: Calculate the new f(x, y) for the updated values of x and y.
3.5: Calculate the trapezoidal increment using the previous and current values of f(x, y), and multiply it by h / 2. The trapezoidal increment is given by (h * (f(x, y) + new f(x, y))) / 2.
3.6: Update the value of y using the trapezoidal increment. Set y = y + trapezoidal increment.
Step 4: Repeat steps 3.1 to 3.6 for the remaining 10 steps, until we reach x = 1.1.
Step 5: Round the final value of y to 4 decimal places to obtain the approximate value of y(1.1).
By following this process, you will be able to calculate the approximate value of y(1.1) using the Euler-trapezoidal method with a step-size h = 0.1.