Consider the differential equation

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

Problem

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

. A lattice-based slope field has been constructed on the rectangle

0\le x\le1

0\le y\le1.5

at grid points spaced

0.1

unit apart in both directions. A solution passes through

(0,1)

. (a) Using only the slope field information, create a table that records successive approximations of

y

x=0.2,0.4,0.6,0.8,\text{ and }1.0

obtained by moving in steps of

\Delta x=0.2

along the slope segments that originate from each left-endpoint. Show the computed change in

y

for each interval. (b) Repeat the process in part (a) with a refined step size of

\Delta x=0.1

. Present both tables side by side and clearly indicate the new approximations of

y(1)

. (c) Using the two approximations of

y(1)

obtained in parts (a) and (b), propose a bound on the true value of

y(1)

. Justify your bound by discussing how the direction and concavity information visible in the slope field affects the over- or underestimation at each step. (d) Explain how decreasing the step size further would influence the accuracy of the estimate and the width of the bound you stated in part (c).

Related theory

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