 
Summary: HIGHER ORDER DIFFERENTIAL EQUATIONS
1. Higher Order Equations
Consider the differential equation
(1) y(n)
(x) = f(x, y(x), y (x), . . . , y(n1)
(x)).
1.1. The Existence and Uniqueness Theorem
Suppose x0 is a given "initial point" x = x0, and suppose a0, a1, . . . , an1 are
given constants. Then there is exactly one solution to the differential equation (1)
which satisfies the initial conditions
(2) y(x0) = a0, y (x0) = a1, y (x0) = a2, . . . , y(n1)
(x0) = an1.
Note that for an nth
order equation we can prescribe exactly n initial values. The
proof of this theorem is difficult, and not part of math 320.
1.2. The general solution
If you try to solve the differential equation (1), and if everything goes well, then
you will end up with a formula for the solution
y = y(x, c1, c2, . . . , cn)
which contains a number of constants. Often the way you got the solution leaves
