Let’s suppose we wish to solve a differential problem of the form:
But we cannot do the separable equations, and also
, so we cannot use exact equations.
But what if we could multiply the whole differential equation by some new equation that would make this problem exact? Let’s suppose there exists some sort of function that can do this trick. This function could be a function of
or possibly some function of
. Let’s suppose that the function that does this trick is
And the whole goal of this is to have
, so we can use our Exact Equations Method.
So if we can choose a
then choose this
and multiply the original equation by it:
Now just solve this using the Exact Equation Method.
As we multiplied the entire equation by
every solution to
will also be a solution to
Which was our original problem.
e.g. If we had an equation
and multiplied the whole equation by
(which could be our
, then we will have
is a solution to both
. The only extra solution we picked up was
, which is the case where