Understanding Engineering Mathematics

• The solution obtained is, in this case, themost general solution–there
  are no others.


• To fixAwe would have to specify an extra condition ony–suchas
  its ‘initial value’ – its value atx=0. If this isy 0 then:

y 0 =Ae^0 =A,soA=y 0

and the solution is

y=y 0 ex

Exercise on 15.1

Solve the more general equation (15.1) in the case when the initial number of bacteria is
n 0 =4. If time is measured in seconds, and after 5 seconds it is found that the number of
bacteria is 10, what is the value ofk?

Answer

n= 4 ekt,k=^15 ln( 5 / 2 )

15.2 Definitions

Anordinary differential equation(DE) for adependent variabley in terms of an
independent variablexis any equation that contains one or more derivatives ofywith
respect tox, and possiblyxandy. The order of the highest order derivative in the equation
defines theorderof the DE.
A DE in which the only power to whichyor any of its derivatives occurs is zero or
one is called alinear DE. Any other DE is said to benonlinear. Except for some special
cases nonlinear equations are very difficult to deal with. Often, particularly in engineering,
they are solved by approximating them by linear equations, although there are many cases
where the full nonlinearity must be confronted. Most of this chapter will be devoted to
linear equations.

Problem 15.2

State the order of each DE, and state which are nonlinear.

(i)

dy

dx

=x^2 (ii)

(

dy

dx

) 2

Yy=x

(iii)

d^2 y

dx^2

Y 4 y=0(iv)

d^2 y

dx^2

Y 4 y^2 = 0

(i) is first order and linear
(ii) is first order and nonlinear
(iii) is second order and linear
(iv) is second order and nonlinear, because of they^2.

Asolutionof a DE foryin terms ofxis any function,y=f(x), which when substituted
into the equation reduces it to an identity – that is, it satisfies it identically, for all values
ofx(50
➤
). We then sayf(x)satisfiesthe equation. In general a DE can have more
than one such solution. Sometimes a single function can be found which incorporates