Maximum or minimum of a function.

MIT OpenCourseWarehttp://ocw.mit.edu 18.034 Honors Differential Equations Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. LECTURE 10. MAXIMUM PRINCIPLE By considering the points where a function attains a maximum or minimum one can get a good deal of information about the solutions of a differential equation without solving it. We begin with a review of terminology. A point x is interior to an interval I if x is in I but is not an endpoint. An interval is bounded if its length is finite and closed if it contains its endpoints. For example, 1 1 is neither bounded nor closed. A real-valued function f(x) defined on the interval I of x is said to have a maximum at x0 if x0 ∈ I and f(x0) � f(x) on x ∈ I. The maximum is called interior if x0 is an interior point of I and it is called positive if f(x0) > 0. The “negative interior minimum” is defined similarly. If the inequality f(x0) � f(x) is required to hold only in some open interval J ⊂ I, with x ∈ J and x0 ∈ J then the maximum is said to be local. In what follows, we use the terms maximum and minimum to mean local maximum and local minimum, respectively. We recall the following theorem of calculus. Theorem 10.1. A continuous real-valued function on a bounded closed interval attains its maximum and minimum on the interval. At an interior maximum or minimum x0 a differentiable function f satisfies f�(x0)=0. If f is twice differentiable then f satisfies the additional condition f��(x0) � 0 at a maximum and f��(x0) � 0 at a minimum. Thus, at a positive interior maximum a twice-differentiable function f satisfies (10.1) f(x0) > 0,f�(x0)=0, and f��(x0) � 0. At a negative interior minimum it satisfies (10.2) f(x0) < 0,f�(x0)=0, and f��(x0) � 0. Our aim here is to show how (10.1) and (10.2) are used to obtain information about the solutions of differential equations. We adopt the convention that a 0 on an interval then y is strictly increasing. The relation y� > 0 is an example of a differential inequality. Differential inequalities form a major subfield of the modern theory of differential equations. Example 10.4. Show that a function y satisfying x ey�� + y� sin x − (1 + x)y � 0, for x> 0 and y(0) � 0, y�(0) > 0 must be strictly increasing. SOLUTION. If otherwise, there are points x1 and x2 such that 0 0 and y(0) = y(b)=0 where b> 0. Show that b>π unless y is identically zero. SOLUTION. Suppose that b � π and that y is not identically zero for 0 0 and λ< 0. 2. Consider e cos x w�� − x 2 w + x 3 =0, for all x, where w(0) = 0. Show that y = w − x have a positive maximum or negative minimum at any value x = c. Show also that the sign of w�� is the same as the sign of w − x. 3. Show that the solution of the DE (cosh x)y�� + (cos x)y� = (1+ x2)y for a