📂Odinary Differential Equations

Linearity of Laplace Transform

Theorem¹

Let $f_{1}$ and $f_2$ be functions for which the Laplace transform exists. Also, let $c_{1}, c_2$ be an arbitrary constant. Then

$$ \mathcal{L} \left\{ c_{1}f_{1} + c_2f_2 \right\} = c_{1}\mathcal{L} \left\{f_{1} \right\} + c_2\mathcal{L} \left\{f_2 \right\} $$

Explanation

It is obvious that the Laplace transform is an integral transform.

Proof

$$ \begin{align*} \mathcal{L} \left\{ c_{1}f_{1}+c_2f_2 \right\} &= \int_{0}^\infty e^{-st} \left( c_{1}f_{1}+c_2f_2 \right) dt \\ &= \int_{0}^\infty e^{-st}c_{1}f_{1} dt + \int _{0}^\infty e^{-st}c_2f_2 dt \\ &= c_{1}\int_{0}^\infty e^{-st}f_{1} dt + c_2\int _{0}^\infty e^{-st}f_2 dt \\ &= c_{1}\mathcal{ L} \left\{ f_{1} \right\} + c_2 \mathcal{ L} \left\{ f_2 \right\} \end{align*} $$

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