Consider the differential equation with initial condition

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

Problem

Consider the differential equation

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

with initial condition

y(0)=\dfrac{\pi}{2}

. A slope field is to be drawn for

0\le x\le2

and

0\le y\le\pi

. (a) Without sketching, explain why the slope field along the horizontal line

y=\dfrac{\pi}{2}

consists of line segments of slope

x

. (b) Construct a table of estimated solution values by advancing in steps of

\Delta x=0.25

from

x=0

x=2

. For each step, use the slope given by the field at the left-endpoint of the subinterval to approximate the increment in

y

. Record all intermediate computations. (c) Using the ordered pairs obtained in part (b), plot an approximate solution curve on graph paper. Discuss the consistency of your plotted curve with qualitative features you would expect from the slope field. (d) Use the approximated values of

y

to estimate

\displaystyle \int_{0}^{2} y\,dx

by applying the trapezoidal rule with the same subintervals. State your result to three decimal places and comment on whether you expect your estimate to be an overestimate or an underestimate, citing evidence from the slope field.

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