Editing Differential equation (section)

==Existence of solutions==

[[Equation solving|Solving]] differential equations is not like solving [[algebraic equations]]. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest.

For first order initial value problems, the [[Peano existence theorem]] gives one set of circumstances in which a solution exists. Given any point <math>(a, b)</math> in the xy-plane, define some rectangular region <math>Z</math>, such that <math>Z = [l, m]\times[n, p]</math> and <math>(a, b)</math> is in the interior of <math>Z</math>. If we are given a differential equation <math display="inline">\frac{dy}{dx} = g(x, y)</math> and the condition that <math>y = b</math> when <math>x = a</math>, then there is locally a solution to this problem if <math>g(x, y)</math> and <math display="inline">\frac{\partial g}{\partial x}</math> are both continuous on <math>Z</math>. This solution exists on some interval with its center at <math>a</math>. The solution may not be unique. (See [[Ordinary differential equation]] for other results.)

However, this only helps us with first order [[initial value problem]]s. Suppose we had a linear initial value problem of the nth order:
:<math>f_{n}(x)\frac{d^n y}{dx^n} + \cdots + f_{1}(x)\frac{d y}{dx} + f_{0}(x)y = g(x)</math>

such that
:<math>\begin{align}
    y(x_{0}) &= y_{0}, &
   y'(x_{0}) &= y'_{0}, &
  y''(x_{0}) &= y''_{0}, &
      \ldots
\end{align}</math>

For any nonzero <math>f_{n}(x)</math>, if <math>\{f_{0},f_{1},\ldots\}</math> and <math>g</math> are continuous on some interval containing <math>x_{0}</math>, <math>y</math> exists and is unique.<ref>{{cite book|last1=Zill|first1=Dennis G.|title=A First Course in Differential Equations|publisher=Brooks/Cole|isbn=0-534-37388-7|edition=5th|year=2001}}</ref>