Ordinary differential equation: Encyclopedia II - Ordinary differential equation - Existence and nature of solutions

Ordinary differential equation - Existence and nature of solutions

The problem of solving a differential equation is to find the function y whose derivatives satisfy the equation. For example, the differential equation

has the general solution

where A, B are constants determined from boundary conditions.

In general, an n-th order equation allows both x and y to be fixed, as well as all the n − 1 lower order derivatives of y; the remaining equation can be solved (at least conceptionally) for y⁽ⁿ⁾. If the equation has finite degree d, then we now have a polynomial equation in y⁽ⁿ⁾ with at most d roots. Therefore there can be as many as d possible values for y⁽ⁿ⁾ at any given point and for any possible values of the lower order derivatives, though there may be ranges of these points and values where there are fewer solutions (or none at all). A Lipschitz condition must also be satisfied for a solution to exist.

Thus, in the previous example, a second-order, first-degree equation, any point on the plane and any slope through that point can be selected and yield a unique solution (since the single root of y'' exists for any value of y). Note in particular that there are an infinity of solutions through any given point; this is a general characteristic of equations of order higher than one.

Consider now

with general solution

This is a first-order, second-degree equation, thus any point can have at most two solutions passing through it, corresponding to the two roots of y' in the quadratic equation that would result after fixing x and y. Studying the quadratic equation's discriminant (x² + 4y) leads to the conclusion that only a single solution exists along the parabola (where the discriminant is zero) and that no solution exists below this parabola (where both roots are complex).

The parabola in this problem is an example of a cusp locus; a curve along which two or more roots of the differential equation are identical. Along such a locus it is possible to move from one general solution to another while still obeying the differential equation; thus the presence of cusp loci introduce the possibility of singular solutions. In this example, the parabola is such a singular solution; it satisfies the original differential equation, and a full set of solutions must include such possibilities as the hybrid solution:

where the cusp locus has been used to connect two particular solutions; note that the first derivative (the only derivative to appear in the differential equation) is continuous at the transitions.

(Johnson, Chapter 5)

Adapted from the Wikipedia article "Existence and nature of solutions", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki