Consider the differential equation dy dx = x+2

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

Problem

Consider the differential equation

\frac{dy}{dx}=\sin x+2.

One particular solution, modeling a certain oscillatory signal, passes through

(\pi,4)

and is written in integral form as

y_{1}(x)=4+\int_{\pi}^{x}\bigl(\sin t+2\bigr)\,dt.

A second instrument following the same law is observed to pass through

(\pi,1)

. (a) Determine an explicit formula for the second solution

y_{2}(x)

in terms of an integral with lower limit

\pi

. (b) Find

y_{2}(0)

and

y_{2}(2\pi)

. Interpret the values in the context of the instrument’s signal relative to

y_{1}(x)

. (c) Show that

y_{2}(x)=y_{1}(x)-3

for all real

x

, and explain geometrically why the two solution curves never intersect. (d) Discuss the physical meaning of the vertical translation between

y_{1}

and

y_{2}

, and describe how any other initial value at

x=\pi

would alter the signal’s graph.