The simplest first order differential equation is of the form
(1). We can solve this equation by assuming a solution of the form
so that
Substituting these into the differential equation (1) gives
since that would mean the solution is trivial (and does not satisfy any boundary conditions) and
so
If
when
then
If instead we have a first order differential equation with vectors, say
(2) with
when
we can still assume a solution of the form
but now
is also a vector
which we have to find given the initial conditions.
If
then
Substituting into the differential equation (2) gives
As before, we can factorise with
to give
Again as before
and
so
hence
(3)
To finduse the initial conditionswhen
hence
so
We can write this solution in vector form as
More complicated equations may include constant terms added. For example
(4)
with
when
We find the solution in two parts. The first part will be the solution to
This is (3) above. The second part is any solution to (4). We can see that if
is a constant then
so put
into (4) then
The general solution is the sum of
and
so
We can find using the initial conditionswhen
Hence
We can write this as