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Use Euler's method with step size 0.1 to estimate y(2.5), where y(x) is the solution of the initial-value problem y' = 3y + 2xy, y(2) = 1. (Round your answer to four decimal places.)

Question

Use Euler's method with step size 0.1 to estimate y(2.5), where y(x) is the solution of the initial-value problem y' = 3y + 2xy, y(2) = 1. (Round your answer to four decimal places.)

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Solution

Sure, let's use Euler's method to solve this problem. Euler's method is a simple numerical method to solve first order, first degree differential equations with a given initial value. Here are the steps:

  1. First, let's define the function for the derivative, which is given as y' = 3y + 2xy.

  2. The initial condition is given as y(2) = 1. So, our starting point is x = 2, y = 1.

  3. The step size is given as 0.1. So, we will increment x by 0.1 in each step.

  4. We need to find the value of y at x = 2.5. So, we will perform 5 steps, because (2.5 - 2) / 0.1 = 5.

  5. In each step, we will use the formula for Euler's method to calculate the new y:

    y_new = y_old + step_size * derivative(x_old, y_old)

    where derivative(x, y) is the function we defined in step 1.

  6. After 5 steps, we will have the estimated value for y(2.5).

Let's perform the calculations:

Step 1: x = 2, y = 1 Derivative = 31 + 22*1 = 7 y_new = 1 + 0.1 * 7 = 1.7

Step 2: x = 2.1, y = 1.7 Derivative = 31.7 + 22.1*1.7 = 12.87 y_new = 1.7 + 0.1 * 12.87 = 2.987

Step 3: x = 2.2, y = 2.987 Derivative = 32.987 + 22.2*2.987 = 22.5794 y_new = 2.987 + 0.1 * 22.5794 = 5.24594

Step 4: x = 2.3, y = 5.24594 Derivative = 35.24594 + 22.3*5.24594 = 39.74382 y_new = 5.24594 + 0.1 * 39.74382 = 9.22932

Step 5: x = 2.4, y = 9.22932 Derivative = 39.22932 + 22.4*9.22932 = 70.37484 y_new = 9.22932 + 0.1 * 70.37484 = 16.26680

So, the estimated value for y(2.5) is 16.26680, rounded to four decimal places.

This problem has been solved

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