MATH 255 Lecture Notes - Lecture 5: 2Wo, Linear Independence, Linear Combination
Document Summary
Week 5 (i) second-order linear odes have the following general form y. + q(x)y = f (x); (1) where p(x); q(x); f (x) are given continuous functions. The initial value problem associated with the second-order odes is to nd solution of. Y y(0) = a (0) = b: y. Similar to the existence and uniqueness theorem for the rst-order odes, there is an existence and uniqueness theorem for (2). That is, given linear equation (1) and a; b r, there is a unique function y(x) which satis es (2). + q(x)y = 0 (3) are simpler situation of (1). 2(0)) are two linearly inde- pendent vectors in r2, if y1(x) and y2(x) are linearly independent solutions of (3). Indeed, if (y1(0); y c1; c2 = 0 such that. 1(0)) and (y2(0); y c1y1(0) + c2y2(0) = 0; c1y. But y(x) := c1y1(x) + c2y2(x) is also a solution of (3), by theorem 2. 1. 1.
