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MIT18_03S10_lt

# MIT18_03S10_lt - LT Laplace Transform 1 Translation formula...

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± LT. Laplace Transform 1. Translation formula. The usual L.T. formula for translation on the t -axis is (1) L u ( t a ) f ( t a ) = e as F ( s ) , where F ( s ) = L f ( t ) , a > 0 . This formula is useful for computing the inverse Laplace transform of e as F ( s ), for example. On the other hand, as written above it is not immediately applicable to computing the L.T. of functions having the form u ( t a ) f ( t ). For this you should use instead this form of (1): (2) L u ( t a ) f ( t ) = e as L f ( t + a ) , a > 0 . Example 1. Calculate the Laplace transform of u ( t 1)( t 2 + 2 t ). Solution. Here f ( t ) = t 2 + 2 t , so (check this!) f ( t + 1) = t 2 + 4 t + 3. So by (2), 2 4 3 s s L u ( t 1)( t 2 + 2 t ) = e L ( t 2 + 4 t + 3) = e 3 + 2 + . s s s Example 2. Find L u ( t 2 ) sin t . Solution. L u ( t = e �s/ 2 L sin( t + 2 2 ) sin t = e �s/ 2 L (cos t ) = e �s/ 2 s .

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MIT18_03S10_lt - LT Laplace Transform 1 Translation formula...

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