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a relation between a function and it's second derivative

Source: Iran PPCE 2012- Analysis exam-P5

February 14, 2012
functioncalculusderivativetrigonometrysearchreal analysisreal analysis unsolved

Problem Statement

The 2nd2^{nd} order differentiable function f:R⟶Rf:\mathbb R \longrightarrow \mathbb R is in such a way that for every x∈Rx\in \mathbb R we have f′′(x)+f(x)=0f''(x)+f(x)=0.
a) Prove that if in addition, f(0)=f′(0)=0f(0)=f'(0)=0, then f≡0f\equiv 0.
b) Use the previous part to show that there exist a,b∈Ra,b\in \mathbb R such that f(x)=asin⁡x+bcos⁡xf(x)=a\sin x+b\cos x.
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