Find approximate values?

Question

Find approximate values of the solution of y'=3+t-y, y(0)=1 at t=0.1, 0.2, 0.3, and 0.4 using the Euler method with h=0.1.

Answer 1

y'(0) = 3 + 0 - y(0) = 2

y(0.1) ≈ y(0) + y'(0)*0.1 = 1.2

y'(0.1) ≈ 3 + 0.1 - y(0.1) ≈ 1.9

y(0.2) ≈ y(0.1) + y'(0.1)*0.1 ≈ 1.39

y'(0.2) ≈ 3 + 0.2 - y(0.2) ≈ 1.81

y(0.3) ≈ y(0.2) + y'(0.2)*0.1 ≈ 1.571

y'(0.3) ≈ 3 + 0.3 - y(0.3) ≈ 1.729

y(0.4) ≈ y(0.3) + y'(0.3)*0.1 ≈ 1.7439

Answer 2

By Euler method,

y(t_n+1) = y(t_n) + y'(t_n) h

y(0) =  1

Do iteration,

y(0.1) = y(0) + y'(0) * h = 1 + (3+0-1) * 0.1 = 1.2

y(0.2) = y(0.1) + y'(0.1) * h = 1.2 + (3+0.1-1.2) * 0.1 = 1.39

y(0.3) = ... = 1.571

y(0.4) = ... = 1.7439