Existence and Uniqueness
Let R be a rectangular region in the x-y plane defined by
that contains the point (x0 , y0) in its interior.
If f(x,y)
and δf/δy are continuous on R,
then there exists some interval I0: (x0-h, x0+h), h > 0,
contained in [a,b], and a
unique function y(x), defined on I0, that is a solution of the initial
value problem
dy/dx = f(x,y)
y(x0) = y0.
What if the equation is linear?
Given the differential equation
an(x) dny/dxn + an-1(x) dn-1y/dxn-1 ... + a1(x) dy/dx + a0(x) y = g(x)
subject to:
y(x0) = y0, y'(x0) = y1,
y"(x0) = y2, ... y(n-1) (x0) = yn-1.
Let an(x) , an-1(x) ... a0(x) and
g(x) be continuous and an(x)/= 0 for every x in an interval I.
If x0 is a point in I, then a unique solution exists on the interval.