$$ L \{ \frac { \partial y } { \partial t } \} = S L \{ y \} - y ( 0 ) $$

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The Laplace transform of the derivative of a function $$ y(t) $$ is $$ s $$ times the Laplace transform of $$ y $$ minus the initial value of $$ y $$ at $$ t = 0 $$. The equation you've provided is related to the Laplace transform of a derivative. The Laplace transform is a powerful tool used in engineering and mathematics to analyze linear time-invariant systems, among other applications. The equation can be interpreted as follows: - $$ L \{ \frac{\partial y}{\partial t} \} $$ represents the Laplace transform of the derivative of a function $$ y(t) $$ with respect to time $$ t $$. - $$ S $$ is the complex frequency variable in the Laplace domain, often denoted as $$ s $$. - $$ L \{ y \} $$ is the Laplace transform of the function $$ y(t) $$. - $$ y(0) $$ is the initial value of the function $$ y(t) $$ at time $$ t = 0 $$. The formula states that the Laplace transform of the derivative of a function is equal to the product of the complex frequency variable $$ s $$ and the Laplace transform of the function, minus the initial value of the function at $$ t = 0 $$. In mathematical terms, the relationship can be expressed as: $$ L \left\{ \frac{\partial y}{\partial t} \right\} = s L \{ y \} - y(0) $$ This formula is very useful when solving differential equations using the Laplace transform, as it allows you to transform the differential equation into an algebraic equation in the Laplace domain, which is often easier to solve. After solving for $$ L \{ y \} $$, you can then apply the inverse Laplace transform to find $$ y(t) $$.

\( L \{ \frac { \partial y } { \partial t } \} = S L \{ y \} - y ( 0 ) \)

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\( L \{ \frac { \partial y } { \partial t } \} = S L \{ y \} - y ( 0 ) \)

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