The method of separation of variables applies to differential equations of the form
where and
are functions of a single variable.
Find the general solution to the differential equation
Any constant solution to this equation would have
so that
.
Avoiding the constant solution, we may divide both sides of the
equation by and then we solve:
So, if we set , we have
.
To solve the differential equation
:
Find the general solution of
We begin by rewriting the equation at
.
The only constant solution is .
Integrating, we find that is an antiderivative of
while
is an antiderivative of
.
Let . Then we have
.
Adding to both sides and applying the exponential function,
we conclude that
.
As the solution must be continuous, the signs of
and
agree. Thus,
.
Note: In this case the constant solution has the same form.
Find the general solution to the differential equation
The method of separation of variables does not apply as the function
cannot be written as the product of a function of
by a function of
.
Scholium: Using Taylor series expansions (a topic which
we shall discuss next month), one can compute an expression
for solutions to the equation .
Find the general solution to the equation
There are no constant solutions as
is never zero.
Note, however, that we cannot have
as the differential
equation would require
to be nondifferentiable at such a point.
As before, we set . Multiplying by
and integrating,
we find
So, satisfies the equation
From the quadratic formula, we compute that
Find a function satisfying
and
.
As the exponential function never attains the value zero, there are
no constant solutions to this differential equation. Multiplying both
sides of the equation by and integrating, we obtain:
Addding to both sides of this equation and taking the natural logarithm, we
compute