Use Euler's Method with three equal steps and y(1) = 1 to approximate y(1.3) for the differential equation dy/dx= x+y^2

a) 1.283
b) 1.695
c) 1.785
d) 2.943

well, you take steps of ∆x = 0.1

Then you have the next y-step is ∆y = ∆x * dy/dx, using the tangent line as an approximation.
That gives you the next y-value.
Do it 3 times. What do you get?
It might help to make a table:

x.....y.....y'.....∆x.....∆y
1.....1.....2.....0.1 ...0.2
1.1 1.2 2.54 0.1 ....0.254
and so on

x......y........y'......∆x......∆y

1.2 1.3 2.89 0.1 0.28
1.3 1.5 3.55 0.1 0.35
1.4 1.65 4.12 0.1 0.41
I think is b, is that correct?

To use Euler's Method, we need to divide the interval [1, 1.3] into three equal steps. This means we need to find the step size, h, where h = (1.3 - 1) / 3 = 0.1.

Now let's proceed with the calculations step by step:

Step 1: First, we need to find the value of f(x, y), which is given by the given differential equation dy/dx = x + y^2.
For the initial condition y(1) = 1, we have f(1, 1) = 1 + 1^2 = 2.

Step 2: Next, we use Euler's method to approximate the value of y at x = 1.1:
y1 = y0 + h * f(x0, y0)
Here, y0 = 1, x0 = 1, h = 0.1, and f(x0, y0) = 2:
y1 = 1 + 0.1 * 2 = 1.2

Step 3: Now, we use Euler's method again to approximate the value of y at x = 1.2:
y2 = y1 + h * f(x1, y1)
Here, y1 = 1.2, x1 = 1.1, h = 0.1, and we need to find f(x1, y1):
f(1.1, 1.2) = 1.1 + (1.2)^2 = 2.44
y2 = 1.2 + 0.1 * 2.44 = 1.444

Step 4: Finally, we use Euler's method one more time to approximate the value of y at x = 1.3:
y3 = y2 + h * f(x2, y2)
Here, y2 = 1.444, x2 = 1.2, h = 0.1, and we need to find f(x2, y2):
f(1.2, 1.444) = 1.2 + (1.444)^2 = 3.085936
y3 = 1.444 + 0.1 * 3.085936 = 1.752594

Therefore, the approximate value of y(1.3) using Euler's method with three equal steps is approximately 1.752594.

None of the given options (a) 1.283, b) 1.695, c) 1.785, d) 2.943) matches the calculated value of 1.752594.

No. To complete your table,

x......y........y'......∆x......∆y
1.00 1.00 2.00 0.10 0.20
1.10 1.20 2.54 0.10 0.254
1.20 1.454 3.314 0.1 0.331
1.30 1.785
So, (C)