Span of Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x is preserved under Differentiation wrt x

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Theorem

Let $a, b, x, \alpha_1, \alpha_2 \in \R$ be real numbers.

Denote:

$f_1 := \map \exp {a x} \map \cos {b x}$
$f_2 := \map \exp {a x} \map \sin {b x}$

Let $\map \CC \R$ be the space of continuous real-valued functions.

Let $\tuple {\map {\CC^1} \R, +, \, \cdot \,}_\R$ be the vector space of continuously differentiable real-valued functions.

Let $S = \set {f_1, f_2} \subset \map {\CC^1} \R$ be a vector space.

Let $\map \span S$ be the span of $S$.


Then differentiation with respect to $x$ preserves $\map \span S$.


Proof

\(\ds \map {\dfrac \d {\d x} } {\alpha_1 f_1 + \alpha_2 f_2}\) \(=\) \(\ds \map {\dfrac \d {\d x} } {\alpha_1 \map \exp {a x} \map \cos {b x} + \alpha_2 \map \exp {a x} \map \sin {b x} }\)
\(\ds \) \(=\) \(\ds \alpha_1 a \map \exp {a x} \map \cos {b x} - \alpha_1 b \map \exp {a x} \map \sin {b x} + \alpha_2 a \map \exp {a x} \map \sin {b x} + \alpha_2 b \map \exp {a x} \map \cos {b x}\)
\(\ds \) \(=\) \(\ds \paren {\alpha_1 a + \alpha_2 b} \map \exp {a x} \map \cos {b x} + \paren {\alpha_2 a - \alpha_1 b} \map \exp {a x} \map \sin {b x}\)
\(\ds \) \(=\) \(\ds \beta_1 f_1 + \beta_2 f_2\)

$\blacksquare$




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