University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2021
Views:

214

Pages:

1
Author:

Daniel Deleon

Summary: 4.1 Here is what we need to know: Given a linear operator L for a differential equation (assume nth order): • If y1 , . . . , yk are solutions to L(y) = 0, so is any linear combination, c1 y1 +. . .+ck yk . (These functions are in the kernel of L). • There are exactly n linearly independent solutions to L(y) = 0. They form the “fundamental set” in that any solution to the IVP can be written in terms of these. • The general solution to L(y) = f1 (x) + . . . + fk (x) is: yh (x) + yp1 + . . . + ypk , where yh is the general solution to L(y) = 0, and ypi (x) solves L(y) = fi (x). In particular, here is what happens when n = 2: 1. The Existence and Uniqueness Theorem. Let y 00 + p(t)y 0 + q(t)y = f (t), y(t0 ) = y0 . Then if p, q, f are all continuous for t ∈ (a, b), (and of course, t0 ∈ (a, b)), then there exists a unique solution to the differential equation, and this solution persists for all t ∈ (a, b). 2. Definitions: (a) Linear Independence: A set of functions, {y1 (t), . . . , yk (t)} is said to be linearly independent on the interval [a, b] iff the only solution to: c1 y1 (t) + . . . + ck yk (t) = 0 (for all t ∈ [a, b]) is: c1 = c2 = . . . = ck = 0 (b) The Wronskian of y1 and y2 is: W (y1 , y2 )(t) = y1 y10 y2 y20 = y1 y20 − y10 y2 Note that this is a function of t. (c) The Fundamental Set of Solutions: y1 and y2 form a fundamental set of solutions to y 00 + p(t)y 0 + q(t)y = 0 if the solution to an aribtrary initial value problem can be written as a linear combination of y1 and y2 . Thus, ALL solutions are of the form: c1 y1 + c2 y2 if y1 and y2 form a fundamental set. 3. Theorem (Linear Independence and the Wronskian, in General) If f, g are differentiable on an open interval I and if W (f, g)(t0 ) 6= 0 for some t0 in I, then f and g are linearly independent on I. However, it is possible in this general case, that the Wronskian is zero for all t in I, and the functions f and g are linearly independent. This is not true in the special case that f, g are two solutions to the second order linear, homogeneous differential equation: 4. Abel’s Theorem (Linear Independence and the Wronskian, for Solutions to a Linear Homogeneous Differential Equation). Let y 00 + p(t)y 0 + q(t)y = 0, and let I be an open interval on which p and q are continuous. Let y1 and y2 be solutions. Then the Wronskian of y1 and y2 is either identically zero on I, (in which case y1 , y2 are linearly dependent) or it is never zero on I (in which case y1 , y2 are linearly independent). 1

Differential Equations Summary 4.1

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