Understanding Engineering Mathematics

all possible solutions of the DE – this is then referred to as thegeneral solution.Such
solutions contain one or more arbitrary constants, usually depending on the order of the
DE. Any other solution than the general solution is called aparticular solution. Particular
solutions can often be found by guesswork (or ‘by inspection’), but general solutions are
usually much harder to find.
The arbitrary constant(s) occurring in the general solution can be determined by supple-
menting the DE by specified conditions in whichy, or a sufficient number of its derivatives,
is given for some particular value(s) ofx. Such conditions are calledinitialorboundary
conditions.
The difference between initial and boundary conditions can be seen by considering a
projectile such as a shell from a gun. Even if you don’t know the differential equation
that describes its motion under gravity alone, you can perhaps appreciate that you could
specify the motion completely in one of (at least) two ways:

• Specify the position and velocity at the initial point of projec-
  tion –initial conditions


• Specify two separate points on the trajectory – say point of projec-
  tion and the furthest point reached, the landing point –boundary
  conditions

Problem 15.3

Show thaty=

x^3

3

Y1 is a solution of Problem 15.2(i). Can you find any

other solutions?

Ify=

x^3

3

+1then

dy

dx

=x^2 and the DE is clearly satisfied. Therefore

y=

x^3

3

+1 is a solution

You may realise that in factanyfunction of the form

y=

x^3

3

+C

WhereCis an arbitrary constant is also a solution, becauseCis knocked out by the
differentiation.

Problem 15.4

Show that the following are solutions of Problem 15.2(iii):

(a) y=sin 2x (b) y=3cos2x

(c) y=2sin2x−cos 2x

Can you suggest any more solutions? Is 2 sinxa solution?

(a) Differentiating twice we have

d^2 y

dx^2

=−4sin2x=− 4 y