Let be a function that satisfies with

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

Let

y

be a function that satisfies

\dfrac{dy}{dx}=x^2-y

with

y(0)=1

. A student uses the following iterative procedure: • Start at

(x_0,y_0)=(0,1)

. • For each step, increase

x

h=0.25

and increase

y

(\text{slope at the current point})\times h

. The student records intermediate values to three decimal places. (a) Compute the student’s approximation to

y(0.25)

. (b) Using the value from part (a), compute the student’s approximation to

y(0.50)

. (c) The exact solution of the differential equation is

y(x)=x^2-2x+2-\dfrac{1}{e^{x}}

. Calculate the exact values of

y(0.25)

and

y(0.50)

, then determine the numerical error in each of the approximations from parts (a) and (b). (d) Compare the magnitudes of the two errors obtained in part (c). Provide a mathematical explanation for the observed change in error between the first and second step.

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