5 Steps to a 5 AP Calculus BC 2019

More Applications of Definite Integrals 327

(b) Sketch a possible graph for the particular solutiony=f(x) to the differential equation
with the initial conditionf(0)=3.
(c) Find, algebraically, the particular solutiony= f(x) to the differential equation with
the initial conditionf(0)=3.

Solution:

(a) Set up a table of values for

dy

dx

at the 15 given points.

x=− 2 x=− 1 x= 0 x= 1 x= 2

y= 12 1 0− 1 − 2

y= 24 2 0− 2 − 4

y= 36 3 0− 3 − 6

Then sketch the tangents at the various points as shown in Figure 13.5-6.

3

2

1

–2 0–1 1 2

y

x

Figure 13.5-6

(b) Locate the point (0, 3) as indicated in the initial condition. Follow the flow of the field
and sketch the curve as shown Figure 13.5-7.

(c) Step 1: Rewrite

dy

dx

=−xyas

dy

y

=−xdx.

Step 2: Integrate both sides

∫

dy

y

=

∫

−xdxand obtain ln|y|=−

x^2

2

+C.

Step 3: Apply the exponential function to both sides and obtaineln|y|=e−

x 22 +C

.

Step 4: Simplify the equation and gety=

(

e

−x 22 )

(eC)=

eC

e

x 22.

Letk=eCand you havey=

k

e

x 22.