A first order linear differential equation is a differential equation of the form
Solve the differential equation
In this case we can use the method of separation of variables.
If is constant, then
so that
.
Otherwise, we may express the equation as
.
Let
. Integrating with respect to
, we have
(As our solution must be continous and cannot take the value zero,
the signs of and
must agree. So, we may
drop the absolute value bars.)
Exponentiating both sides of this equation and multiplying by , we obtain
.
Solve the differential equation
In this case, we cannot apply the separation of variables technique.
However, as is never equal to zero, the solutions to the original
equation and to the equation
are the same.
Observe that
We integrate with respect to .
So, if we write , then we have
.
Solve the differential equation
In this case, multiplying by we may express the
equation as
. Using the
product rule we check that
.
We integrate this expression.
Note: The original equation
is singular at in the sense that the function
is
not defined. We need to take for the lower limit of integration
some other constant. The number
is a convenient choice in this case.
Write . Then we conclude that
.
In general, if , then
Thus, a differential equation of the form
may be expressed as
.
So, if is in the domain of the functions
and
, we
have
Set
, then
.
In solving the equation
, we multiplied by
and then observed that
.
In terms of the general solution,
and if
, then
we have
.
Note that
. So, multiplying by
is the same
as multiplying by
for
.
Our general method gives
To finish, we must choose and evaluate the above integral.