Isolated Zeros. Let ?(t) be a solution to y’’+py’+qy = 0 on (a, b), where p, q are continuous on (a, b). By completing the following steps, prove that if ? is not identically zero, then its zeros in (a, b) are isolated, i.e., if ?(t0) = 0, then there exists a ? > 0 such that for ?(t) 0 for 0 < |t-t0| < ?.(a) Suppose ?(t0)=0 and assume to the contrary that for each n = 1,2,…, the function ? f has a zero at tn, where 0 < |t0-tn| < 1/n. Show that this implies ?’(t0)=0 [Hint: Consider the difference quotient for ? at t0.](b) With the assumptions of part (a), we have ?(t0)= ?’(t0)=0 Conclude from this that ? must be identically zero, which is a contradiction. Hence, there is some integer n0 such that ?(t) is not zero for 0 < |t-t0| < 1/n0

Introduction to Chapter 1: Difference of Squares: a -b =(a-b)(a+b) Zero Product Property: ab=0; a, b=0 1.1 Linear Equations are first-degree equations in one variable that contain the equality symbol, “=”. Linear equation in the variable x can be expresses in the standard form, ax+b=c, where a ≠, b ∊ ℝ The domain is the largest set of acceptable input values. Of note, the denominator of a fraction cannot be equal to zero. 1.1.1 Solve ax+b=0, solve for x ax+(b-b)=(0-b) 1. Isolate x by subtracting b from both sides. x/a=-b/a 2. Divide both sides by a x=-b/a 1.1.2 Solve 5x-3=2x+8 -2x -2x 1. Subtract 2x from each side 3x-3=8 +3 +3 2. Ad