What is the approximation for f(2) obtained by using Euler's method with two steps of equal length, starting at x=1

Question

Item 6

Item 6

Let y=f(x) be the solution to the differential equation dy/dx=2x+y with initial condition f(1)=0. What is the approximation for f(2) obtained by using Euler's method with two steps of equal length, starting at x=1

Answer 1

f(1.5) ≈ f(1) +0.5 f'(1)

f'(1) = 2x1 + 0 = 2

f(1.5) ≈ 0 + 0.5 x 2 = 1

f(2) ≈ f(1.5) + 0.5 x f'(1.5)

f'(1.5) = 2 x 1.5 + 1 = 4

f(2) ≈ 1 + 0.5 x 4 = 3