Consider the initial value problem with

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

Problem

Consider the initial value problem

\dfrac{dy}{dx}=y-\sin x

with

y(0)=2

. An explicit solution exists, but a student instead produced the numerical estimates of

y(1)

shown in the table below, each obtained with a uniform step size

h

over the interval

0\le x\le 1

. The exact value of

y(1)

2.7182818\ldots

step size $h$	numerical estimate of $y(1)$
$0.200$	$2.420$
$0.100$	$2.569$
$0.050$	$2.644$

Use the information in the table to analyze the relationship between step size and accuracy of the approximation. (a) For each value of

h

in the table, calculate the absolute error

|\text{estimate}-2.7182818|

to six decimal places. (b) Describe the trend you observe in the absolute error when the step size is halved. Support your description with numerical evidence from part (a). (c) Assuming the same trend continues, predict the absolute error that would be expected if the step size is halved once more to

h=0.025

. Explain your reasoning.

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