If x is bigger than zero, then y grows exponentially as x increases.
As x increases, the ratio of f(x) to g(x) is an asymptote approaching zero.
This is a somewhat mysterious mathematical function, written sin(x), and sin(x) will oscillate up and down between 1 and -1 forever as x increases.
If the difference between the solutions approaches zero as x increases, the solution is called asymptotically stable.
All these solutions except y = 1 are stable because they all approach the lines y = 0 or y = 2 as x increases for any values of c that allow the solutions to start out close together.
The solution y = cex of the equation y′ = y, on the other hand, is unstable, because the difference of any two solutions is (c1 - c2)ex, which increases without bound as x increases.
For example, the solution y = ce-x of the equation y′ = -y is asymptotically stable, because the difference of any two solutions c1e-x and c2e-x is (c1 - c2)e-x, which always approaches zero as x increases.
The solution y = 1 is unstable because the difference between this solution and other nearby ones is (1 + c2e-2x -1/2, which increases to 1 as x increases, no matter how close it is initially to the solution y = 1.
In other words, as x increases from a to b, the derivative f′(x) is positive while the function f(x) rises to its maximum value, f′(x) is zero at the value of x for which f(x) has a maximum value, and f′(x) is negative while f(x) declines from its maximum value.
Obvious crystal lattice expansion can be found as x increases.
Hence, predictions from the SDM should become more accurate as x increases.
