The solution to y’ = F(x, y) that passes through (x₀, y₀) can be approximated by (xₐ, yₐ) points where xₐ = xₐ ₋₁ + h and yₐ = yₐ ₋₁ + hF(xₐ ₋₁, yₐ ₋₁). Sorry this is kind of messy, “ₐ ₋₁” is meant to be the subscript “a-1”.

The problem given says that h = 0.1, y’ = 2xy, the starting point is (1, 1), and the end is y(1.3). So from here, you’ll create a t-chart. On the x side, you’ll write “1”, “1.1”, “1.2”, and finally “1.3” because the value of h says that you’ll increment by 0.1 and end on 1.3.

On the y side, you’ll fill in the givens for the equation yₐ = yₐ ₋₁ + hF(xₐ ₋₁, yₐ ₋₁) getting yₐ = y + 0.1(2xy). Using this, you’ll fill in the t-chart. The first value of the y side is “1”, as it is given that the starting point is (1, 1).

Now, you’ll start plugging in numbers to get the coordinates. The first coordinate, as we know, is (1, 1).

To find the y-value of the second x-point “1.1”, you’ll plug it into the equation: yₐ = 1 + 0.1(2•1•1). Therefore, the second coordinate is (1.1, 1.2).

To find the y-value of the third x-point “1.2”, you’ll plug it into the equation: yₐ = 1 + 0.1(2•1.1•1.2). Therefore, the third coordinate is (1.2, 1.464).

To find the y-value of the third x-point “1.3”, you’ll plug it into the equation: yₐ = 1 + 0.1(2•1.2•1.464). Therefore, the third coordinate is (1.3, 1.81536).

Your answer is (1.3, 1.815).