separation of variables
Instead of using differentials, we can also make use of the change of variables theorem for integration and the fact that if two integrable functions are equivalent, then their primitives differ by a constant . Here, we write and for clarity. The above equation then becomes:
Integrating both sides over gives us the desired result:
By the change of variables theorem of integration, the left hand side is equivalent to an integral in the variable :
It follows then that
where is an antiderivative of and is the constant difference between the two primitives. This gives a general form of the solution. An explicit form may be derived by an initial value.
Example: A population that is initially at organisms increases at a rate of each year. We then have a differential equation
The solution of this equation is relatively straightforward, we simply separate the variables algebraically and integrate.
This is just or
When we substitute , we see that . This is where we get the general relation of exponential growth
|Title||separation of variables|
|Date of creation||2013-03-22 12:29:24|
|Last modified on||2013-03-22 12:29:24|
|Last modified by||slider142 (78)|