Understanding Engineering Mathematics

15.3 First order equations – direct integration and

separation of variables

We will only consider cases where we can solve the given DE to give an equation for

dy
dx
of the form

dy

dx

=F(x,y)

whereF(x,y)is a ‘well behaved’ (i.e. we can do whatever we wish with it) function of
xandy.
The ease with which we can solve such a DE will depend on the form ofF(x,y).We
will build up from the simplest cases.
The simplest case is an equation of the form

dy

dx

=f(x)

which can be easily ‘solved’ in principle bydirect integration:

y=

∫

f(x)dx+C

The only difficulty here lies in actually performing the integration. Such simple equations
illustrate many of the key points of DEs in general.
A less trivial variation on this is an equation of the form

dy

dx

=g(y)

We can in fact turn this upside down [not a trivial matter, but permissible with
care (235
➤
)]:

dx

dy

=

1

g(y)

Now this can be integrated directly, with respect toy:

x=

∫

dy

g(y)

+C

This results in principle in a solution of the form

x=G(y)+C

whereG(y)is some function ofy. We may or may not be able to solve this integrated
equation foryin terms ofx.
If we are also given an initial condition, then we can find the value ofCby substituting
this in the result.