The actual solution is the specific solution to a differential equation which not only satisfies the differential equation, although also satisfies the specified initial conditions.

Illustration: What is the actual solution to the subsequent IVP?

2ty' + 4y = 3;    y(1) = -4

Solution: This is in fact easier to do than this might at first appear. From the earlier illustration we already identify that differential equations have all solutions are of the form:

y(t) = 3/4 + c/t²

All that we require to do is find out the value of c that will provide us the solution that we're after. To determine this all we require do is utilize our initial condition that are given as:

-4 = y(1) = 3/4 + c/1²

c= -4 -3/4 = -19/4

Thus, the actual solution to the Initial Value Problem is:

y(t) = ¾ - 19/4t²

From this last illustration we can notice that once we have the general solution to a differential equation determining the actual solution is nothing more than applying the initial conditions and resolving for the constants which are in the general solution.