Question

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 = 9x²,

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 = 3 + e^−x3

is the exact solution of the differential equation.

y = 3 + e^−x3

     ⇒     y' =

LHS = y' + 3x²y =

+ 3x²(3 + e^−x3) = −3x²e^−x3 +

+ 3x²e^−x3 = 9x² = RHS
y(0) =

+ e⁻⁰ = 3 + 1 = 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  ---Select--- divided by 5 multiplied by 10 divided by 10 multiplied by 5 divided by 100 (approximately).

Answer 1

Note: when h = 0.1, some part of calculation is