Let satisfy the differential equation with initial condition

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

Problem

Let

y=g(x)

satisfy the differential equation

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

with initial condition

g(0)=2

. The rectangle

0\le x\le2

0\le y\le2

is shown on a blank set of axes. (a) On the axes, sketch the slope field for the differential equation at each point where

x

and

y

are multiples of 0.5 inside the rectangle. (b) On the same axes, draw a reasonable sketch of the particular solution curve passing through

(0,2)

that remains inside the rectangle. (c) Using your sketch, estimate the value of

g(1.5)

to the nearest tenth. Clearly indicate how you obtained your estimate from the graph. (d) Justify whether

g(x)

can cross the line

y=0

for

0\le x\le2

. Support your answer analytically using the differential equation.