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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?)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 (sinx)' =cosx. (Hint: what can you say about the functions f(x)cosx – f'(x)sinx and f(x)sinx + f'(x)cos 2?)
Transcribed Image Text: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 (sinx)' =cosx. (Hint: what can you say about the functions f(x)cosx – f'(x)sinx and f(x)sinx + f'(x)cos 2?)
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