What kind of math is this 😭 I want to help you (I can learn things very quickly) but I've never done this type of math before

Euler's method uses the derivative to approximate how much the function changes over a small interval.

Since y'(x) = dy/dx, we can estimate the derivative at x = 1 as:

y'(1) ≈ (y(1.5) - y(1)) / (1.5 - 1)

Substituting the given values:

y'(1) ≈ (4 - k) / 0.5

We also know from the differential equation that:

y'(x) = 2x + y

At x = 1, y = k, so:

y'(1) = 2(1) + k = 2 + k

Now set the two expressions equal:

2 + k = (4 - k) / 0.5 = 8 - 2k

3k = 6

k = 2

This is why the solution uses F(1, k) = 2(1) + k. We substitute x = 1 and y = k into the formula y' = 2x + y.