(a) Program a calculator or computer to use Euler's method to compute y(1), where y(x) is the solution of the given initial-value problem. (Give all answers to four decimal places.)

dy

dx

+ 3x²y = 6x²,

y(0) = 4

h = 1	y(1) =
h = 0.1	y(1) =
h = 0.01	y(1) =
h = 0.001	y(1) =

(b) Verify that

y = 2 + 2e^x3

is the exact solution of the differential equation.

y = 2 + 2e^x3

y' =

LHS = y' + 3x²y =

+ 3x²(2 + 2e^x3) = 6x²e^x3 +

+ 6x²e^x3 = 6x² = RHS

y(0) =

+ 2e⁰ = 2 + 2 = 4

(c) Find the errors in using Euler's method to compute y(1) with the step sizes in part (a). (Give all answers to four decimal places.)

h = 1	error = (exact value approximate value) =
h = 0.1	error = (exact value approximate value) =
h = 0.01	error = (exact value approximate value) =
h = 0.001	error = (exact value approximate value) =

What happens to the error when the step size is divided by 10?

When the step size is divided by 10, the error estimate is

(approximately).