Euler's Method....?

Question

Consider the initial-value problem
y' = -y, y(0) = 1.
The analytic solution is: y(x) = e^(-x).

(a) Approximate y(0.1) using one step Euler's method.
(b) Find a bound for the local truncation error in y_1 (where '_' means subscript).
(c) Compare the error in y_1 with your error bound.

Thanks!

Answer 1

(a) y' = -y, so y'(0) = -y(0) = -1.
So for one step Euler's method we get
y_1 = y(0) + y'(0) (y_1 - 0) = 1 + (-1) (0.1) = 0.9.

(b) Error is bounded by (y"(x)/2)(0.1)^2 for some x in [0, 0.1], and y"(x) = e^(-x) so max error bound is (0.1)^2 / 2 = 0.005.

(c) Actual error is e^-0.1 - 0.9 = 0.004837 (4 s.f.) which is 96.7% of the maximum error bound given in (b).