Hailed by The American Mathematical Monthly as `a rigorous and lively introduction,` this text explores a topic of perennial interest in mathematics. The author, a distinguished mathematician and formulator of the Hurewicz theorem, presents a clear and lucid treatment that emphasizes geometric methods. Topics include first-order scalar and vector equations, basic properties of linear vector equations, and two-dimensional nonlinear autonomous systems. Suitable for senior mathematics students, the text begins with an examination of differential equations of the first order in one unknown function. Subsequent chapters address systems of differential equations, linear systems of differential equations, singularities of an autonomous system, and solutions of an autonomous system in the large.
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Differential Equations of the First Order in One Unknown Function
PART A.
THE CAUCHY-EULER APPROXIMATION METHOD
1.Definitions. Direction Fields
By a domain D in the plane we understand a connected open set of points; by a closed domain or region
, such a set plus its boundary points. The most general differential equation of the first order in one unknown function is
where F is a single-valued function in some domain of its arguments. A differentiable function y(x) is a solution if for some interval of x, (x, y(x), yā²(x)) is in the domain of definition of F and if further F(x, y(x), yā²(x)) = 0. We shall in general assume that (1) may be written in the normal form:
where f(x, y) is a continuous function of both its arguments simultaneously in some domain D of the x-y plane. It is known that this reduction may be carried out under certain general conditions. However, if the reduction is impossible, e.g., if for some (x0, y0, y0ā²) for which F(x0, y0, y0ā²) = 0 it is also the case that [āF(x0, y0, y0ā²)]/āyā² = 0, we must for the present omit (1) from consideration.
A solution or integral of (2) over the interval x0 ā¤ x ā¤ x1 is a single-valued function y(x) with a continuous first derivative yā²(x) defined on [x0, x1] such that for x0 ā¤ x ā¤ x1:
Geometrically, we may take (2) as defining a continuous direction field over D; i.e., at each point P: (x, y) of D there is defined a line whose slope is f(x, y); and an integral of (2) is a curve in D, one-valued in x and with a continuously turning tangent, whose tangent at P coincides with the direction at P. Equation (2) does not, however, define the most general direction field possible; for, if R is a bounded region in D, f(x, y) is continuous in R and hence bounded
where M is a positive constant. If Ī± is the angle between the d...
