Understanding Engineering Mathematics

(iii) Discuss the forms of motion described by the relations between position,s,and

time,t, given below.

(a)s=at+b (b)s=at^2 +bt+c

(c)s=asinωt (d)s=ae−bt

In each casea,b,c,ωare all constants which may be positive or negative.

2.Newton’s second law gives us lots of examples ofdifferential equations. You will
study these in some detail in Chapter 15. For now, we just want to get used to what
is meant by asolutionto such an equation. A differential equation is any equation
containing one or more derivatives. Examples are given below, (i) – (vi). A solution to
such an equation is anyfunctionwhich when substituted into the equation, with its
appropriate derivatives, makes the equation identically satisfied – i.e. it produces an
identity in the independent variable. In (a) – (g) are listed a number of functions. By
differentiating and substitution in the equations, determine which functions are solutions
of which equations. Using these examples as inspiration, try to find other solutions of
the equations, which you can then test.
Note that the full exercise of finding solutions to differential equations is at the very least
a matter of integration (Chapter 9), and at its most advanced level may involve powerful
mathematical tools such as transform theory (Chapter 17) or numerical methods.

(i)

dy

dx

+ 3 y= 0 (ii) x

dy

dx

−y=x (iii)

d^2 y

dx^2

+ 4 y= 0

(iv)

d^2 y

dx^2

+ 2

dy

dx

− 3 y=0(v)

d^2 y

dx^2

+ 6

dy

dx

+ 9 y= 0

(vi)

d^2 y

dx^2

+ 2

dy

dx

+ 2 y= 0

(a) 3 cos 2x (b) 2ex (c) − 2 e−xsinx (d) 4e−^3 x

(e) 4x (f) − 6 xe−^3 x (g) xlnx

Each one of the equations (i) – (vi) is an example of a general type of differential

equation that plays a crucial role in one or more areas of engineering science.

3.There are many occasions in engineering when we need to know something about the
tangent and normal to a curve at a given point (230

➤

). For example the reaction of

a force at a smooth surface is normal, i.e. perpendicular, to the surface. In Chapter 5

we looked at tangents to circles – the normal at any point is of course along the radius

(158

➤

). Using differentiation regarded as the slope of a curve, we can find the tangent

at any point on a curve. Also, using the result that the gradients of perpendicular lines

multiply to give−1 (214

➤

), we can then find the normal to a given surface, as the line

perpendicular to the tangent. Investigate the tangents and normals to a general curve at

a given point, using as examples the following:

(i) y=x^2 + 3 x−4at( 0 ,− 4 ) (ii) y=cos 3xatx=

π

4

(iii) y=e^3 xatx=1(iv)y=2lnxatx= 1