To address the issues of existence and uniqueness, consider the simplest form of the first-order ordinary differential equation, namely
with f(t) a given function. A natural and intuitive approach to solving this problem comes from calculus. Referred to as the ``Second Fundamental Theorem of Calculus'' in some calculus texts, equation (5.2) is solved completely from the analytical vantage point through integration:
Using the initial condition 
, the constant of integration is uniquely determined and we obtain the
closed-form formula
That is, analytically our quest at finding the solution is complete,
with the sole caveat that the function f(t) is integrable
in some neighborhood of 
.
In looking to the solution of the more general format of the first-order ordinary differential equation, (5.1), the approach taken is very similar in spirit and the analytical solution of the differential equation becomes a question of judiciously reformulating the problem into an equivalent integration problem. For example, the technique of integrating factors can be used to solve a slightly more general right-hand side of the differential equation than given in (5.2). Consider the ordinary differential equation
Moving terms around, the equivalent problem becomes
The technique of integrating factors is motivated by the question of whether the above equation could be multiplied through by some function - appropriately referred to as the integrating factor - which would cast the left-hand side of the above expression into the derivative of a common expression. Most commonly, this expression takes the form of a derivative of a product of two functions. If such an integrating factor can be found, then solving the expression is as simple as integrating both sides.
In detail, consider the task of finding an integrating factor, v(t), which when multiplies (5.5), transforms the left-hand side of the equation to the derivative of the product v(t)y(t). Thus, such an integrating factor would permit
Thus, integration of equation (5.6) yields
If division by v(t) is possible, i.e. 
, then the analytic derivation of solution y(t) is complete:
Looking back at the motivation of what we were looking for in an integrating factor and the implication it had on equation (5.6), it is clear that in order for these results to hold together, such an integrating factor needs to satisfy the property
This can be seen by observing that equality must hold between the two
left-hand sides of equation (5.6)
above. This is a first-order ordinary differential equation as well, one
that is separable. Again, the condition
that was necessary in obtaining the formula (5.9)
appears here in the separable equation technique for solving v'(t)
=-a(t)v(t). Recall the separation of variables
technique from calculus for solving such an equation, which involves placing
all terms involving v on one side of the equation and all non-v
terms on the other. In the problem at hand, we assume that
so that we can divide through by v(t) and obtain the equation
Using the chain rule, integration of both sides of this expression with respect to t yields
Hence
Now, it's time to wrap up some loose ends in our
above derivation. Our assumption that
here appears to be well-founded inasmuch as the exponential function on
the right-hand side of the above calculation is never zero provided a(t)
is finite. Moreover, the choice of sign ( 
) is ultimately irrelevant due to the nature of the problem, (see equation
5.9). Consequently, the choice of
the integrating factor v(t) is complete:
At the risk of giving the incorrect impression that all differential equations are uniquely solvable, the problem of when unique solutions to equation (5.1) exist should be made precise. The result from the area of differential equations which addresses exactly this point is embodied below.
In Theorem 5.1,
`` 
'' refers to the partial derivative of the function f(t,z)
with respect to the variable z. If partial derivatives are new to
you, a partial derivative of a function with respect to the variable z
is loosely defined as the regular derivative of the function with respect
to the variable z when you consider all other variables (such as
t) to be constants.
To illustrate the ideas presented in Theorem 5.1, we conclude this section with the following example.
