Let f : R → R be a twice differentiable function such that f" + f = 0. Prove there exist constants cl and c2 such that, for all real x, f(x) = c1sinx + c2cosx. We are assuming here that we know all the basic properties of sines and cosines, such as (sinæ)' =cosx. (Hint: what can you say about the functions f(x)cosx – f'(x)sinx and f(x)sinx + f'(x)cos x?)

Let f : R → R be a twice differentiable function such that f" + f = 0. Prove there exist constants cl and c2 such that, for all real x, f(x) = c1sinx + c2cosx. We are assuming here that we know all the basic properties of sines and cosines, such as (sinæ)' =cosx. (Hint: what can you say about the functions f(x)cosx – f'(x)sinx and f(x)sinx + f'(x)cos x?)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.3: Change Of Basis

Problem 12EQ

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