Question

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

Answer 1

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

Answer 2

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?

Answer 3

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)