Let satisfy with

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

Problem

Let

y=y(x)

satisfy

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

with

y(0)=1

. Beginning at

x_0=0

, a variable-length sequence of subintervals is generated according to the following rule: • Start with

h_1=0.20

. • After completing the

k^{\text{th}}

step, compute

|y_k-y_{k-1}|

. If this difference exceeds

0.05

, the next subinterval length is halved; if the difference is less than or equal to

0.05

, the next subinterval length is doubled, but never allowed to exceed

0.20

. Continue the process until

x

first reaches or exceeds

0.60

. (a) Apply the given decision rule to construct a table of

(x_k,h_k,y_k)

values from

k=1

onward, terminating as soon as

x_k\ge0.60

. Show every arithmetic step clearly. (b) State the total number of subintervals required by the adaptive process in part (a). Then determine how many subintervals would have been required if a fixed step length of

0.10

had been used instead. Comment on the relative efficiency of the two approaches.

Related theory

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