sor of the numbers 6975 and 525, we consider one to be the large number and the
other the small number. We already know both numbers are divisible by 0 and 5 (as
they both end in 5), but how do we determine if they have a larger divisor? And if so,
what is that number?
The key is to take the remainders of long division until we arrive at a remainder of
zero. In this instance, we would first divide 6975 by 525; the answer comes out to 13
and a remainder:
Take the remainder—in this case, 150—and then divide it into the 525. Take that
remainder (which turns out to be 75) and divide it into the 150 remainder. The next
iteration leads to no remainder (zero).
Large Number Small Number Remainder
6975 525 150
525 150 75
150 75 0Thus, taking the number before we reached the zero—or 75—gives us the largest
common divisor of both 6975 and 525.
What are systems of differential equations?
In real-life situations, quantities and their rate of change depend on more than one
variable. For example, the rabbit population, though it may be represented by a single
number, depends on the size of predator populations and the availability of food. In
order to represent and study such complicated problems we need to use more than
one dependent variable and more than one equation. Systems of differential equations
are the tools to use. As with the first-order differential equations, the techniques for
studying systems fall into the following three categories: analytic, qualitative, and
numeric.
What is a nonlinear differential equation?
From above, we know that linear equations have specific rules; for example, the
unknowns y, y', and so on, will never be raised to a power more than 1; they will not
be in the denominator of a fraction; ytimes y' is never allowed (because two
- 1575
150
1725525525 697513g