18.03 Class 1, Feb 3, 1999

Introduction. What are differential equations; how they arise; what is a solution to a DE; types of solution (particular, general, implicit); initial conditions; separable equations; orthogonal families.

I’ll try to draw the analogy with an algebraic equation, which typically has one or at worst a finite number of numbers as solutions. Meaning of “ordinary” and “order.”

The simplest is y′ = f(x). Its solution is the subject of “integral calculus,” and much of 18.03 is devoted to reducing more complicated DEs to this one so as to use integration.

I’ll give a simple example (f(x) = x^{2}); note the constant of integration, and that you get a new one for every successive integration, so an n-th order equation should have an n-parameter family of solutions. This is the general solution; a particular solution is typically picked out by specifying an initial condition.

Differential equiations come mostly from scientific models. The science tells us only how a system changes “from one instant to the next.” This is the DE. The mathematical discipline of Differential Equations amplifies this information, giving an analytic or numerical description of the global behavior of the system. Of special interest is the longterm behavior. Once the math is done, the science reenters to interpret this information.

Within mathematics DEs often arise from geometry. For example: what is the increasing curve through (0, 1) with the property that for each (x, y) on the curve the the area of the triangle formed by the x-axis, the tangent line to the curve at (x, y), and the vertical line through (x, y), is equal to 1. This leads to the DE y′ = y^{2}/2.

I’ll solve this DE by separating variables. Notice that the two constants of integration can be merged to one. The resulting curves are horizontal translates of one another, as must be the case since the location of the y-axis does not enter into the statement of the problem. The particular solution containing (0, 1) is y = 2/(2 − x). Note also that the solutions of this DE do not extend over the whole range of values of x for which the DE looks nice: a reflection of the fact that it is a nonlinear DE.

A first order DE gives rise to a family of curves, and, conversely. For example y = ce^{−x} satisfies y′ = −y for any c. The orthogonal family of curves satisfies y′ = 1/y, which has as solution curves the parabolas x = y^{2}/2 + c. While we could solve here for y in terms of x (at the expense of restricting to the top or bottom half of each parabola), it’s more natural to leave the solution as is. More generally, a solution may appear as simply a relation between x and y; this is an implicit solution.