Understanding Engineering Mathematics

of the body, water flow between connected reservoirs, or financial transactions in a

commercial environment. This exercise illustrates the basic ideas in this topic.

Take as our compartment a tank containing 100L of water, into which a brine solution

flows at a rate of 5L/min and out of which solution flows at 5L/min. The concentration

of the incoming brine solution is 1 kg/L. Construct a model, based on reasonable

assumptions, which will enable you to determine the concentration of salt in the tank

at any time after inflow commences.

6.There are many areas of science and engineering where second order linear differential
equations provide useful models. Rather than become enmeshed in the technical details
of specific applications, we will look at generic models which have utility across a
wide range of applications.
The general second order inhomogeneous linear equation with constant coefficients

ax ̈+bx ̇+cx=f(t) ( 15. 6 )

arises naturally in dynamics as a consequence of Newton’s second law

mx ̈=F(t,x,x) ̇

in which the forceF arises from a particular physical set up. For example if we are

talking about the motion of a particle of massmattached to a spring, oscillating in a

resisting medium, and subject to an additional time dependent forcef(t), then we might

model the forces acting as follows, takingxas the displacement from equilibrium:

(i) spring restoring force,−αx α > 0

(ii) resistance force proportional to velocity,−βxβ> ̇ 0

(iii) forcing termf(t)

Newton’s second law then gives

mx ̈+βx ̇+αx=f(t)

A similar equation was mentioned in Chapter 2 (79

➤

) for the capacitor voltage in a

source free electrical circuit containing an inductanceL, resistanceR, capacitanceC.

With an appropriate source termf( 0 )this becomes

d^2 V

dt^2

+

R

L

dV

dt

+

1

LC

V=f(t)

which is, apart from the physical constants, the same as the Newton’s law equation

above. The same sort of equation describes many other systems of widely varying

types – this is mathematical technology transfer!

Such equations are used to describe some type ofoscillatorybehaviour, with some

degree ofdamping. We will concentrate our attention on the general properties of such

behaviour, and adopt the mechanical notation of the first equation above.

The caseβ= 0 =f(t)is the simplest to deal with, yielding unforced, undamped

simple harmonic motion:

mx ̈+αx= 0