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'= -3x+y^2, y(0)=1

To use Euler's method to estimate y(1), we will iteratively calculate the value of y(x) for a sequence of x-values, starting from the initial condition y(0) = 1 and incrementing x by the step size of 0.2.

Here are the steps to follow:

1. Define the step size (h) as 0.2.
2. Set the initial condition: y(0) = 1.
3. Start with x = 0 and y = 1.
4. Iterate the following steps until you reach x = 1:
a. Calculate dy/dx using the given differential equation: dy/dx = -3x + y^2.
b. Calculate the change in y as dy = (dy/dx) * h.
c. Update the value of y: y_new = y + dy.
d. Update the value of x: x = x + h.
5. Repeat steps 4a-4d until x = 1.

Let's go through the iterations:

Iteration 1:
x = 0, y = 1
dy/dx = -3 * 0 + 1^2 = 1
dy = 1 * 0.2 = 0.2
y_new = 1 + 0.2 = 1.2
x_new = 0 + 0.2 = 0.2

Iteration 2:
x = 0.2, y = 1.2
dy/dx = -3 * 0.2 + 1.2^2 = 0.88
dy = 0.88 * 0.2 = 0.176
y_new = 1.2 + 0.176 = 1.376
x_new = 0.2 + 0.2 = 0.4

Continue this process until x reaches 1.

Iteration 6:
x = 0.8, y = 1.6157334832
dy/dx = -3 * 0.8 + 1.6157334832^2 = -0.925655473152
dy = -0.925655473152 * 0.2 = -0.18513109463
y_new = 1.6157334832 - 0.18513109463 = 1.43060238857
x_new = 0.8 + 0.2 = 1

Now, you have obtained an estimate for y(1) using Euler's method with a step size of 0.2. The estimated value is y(1) ≈ 1.43060238857.