When you differentiate a function, the resulting equation has a new function along with a derivative in it. This was demonstrated under Solutions. By integrating this equation you get back the original equation along with a constant of integration. Therefore, every time you perform an integration, you are solving a differential equation.
The result you get is called the General Solution to the differential equation. Given an initial value for x & f(x), you can then solve for the Particular Solution. The following example demonstrates.
Notice how LiveMath has combined the two constants into one and re-labels it. This demonstrates the concept of essential constants discussed under Constants & Number of Solutions Because there are two integrations, two constants are generated. Since these two can be combined into one without effecting the solution, one or the other is non-essential, and can be eliminated.
Looking at it another way, when constants (parameters) enter into an equation as a result of an integration in such a way that different values given to the parameters do not give a different equation family, it is often desirable to reduce, or combine, parameters. For example the simple equation y = x + c1 + c2 is the same equation whether c1 = 3 & c2 = 4 or c1 = 4 & c2 = 3. Both are the same family of lines as the equation y = x + c where c is a single parameter and equals 7.
Since the problem has only one constant now, you can elimate the subscript. This has been done below.
If you are given an Initial Value (IV), you can obtain the Particular Solution by substituting those values into the General Solution and solving for c.
A Graph Theory containing several solution curves including the Particular Solution with the Initial Value, IV (2,3) is displayed below:
First Order Ordinary
Expl #2 A Simple Diffy-Q
