<aside> 🐥 $F(s) = \int ^{\infty} _0 e^{-st} f(t) dt$

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• Used to simplify solution of linear differential equations
• Differential equations and now be transformed into the algebraic equations in the “s-domain”
  • Once solved algebraically, the inverse Laplace transform can be applied to obtain the solution in the time domain

Property 1: Linearity

Property 2: Transformation of a Time Derivative

• Laplace transform of $t^n$ given by

  

Property 3: Translation in the $s$ Domain

Also known as Frequency Shift Theorem:

• Multiplication by an exponential function in the time domain corresponds to a shift in the $s$ domain
• If you multiply a time-domain function by an exponential (can represent growing or decaying oscillations), in the Laplace world, its like you’re shifting the function to a new location

The Inverse:

• Given the Laplace function $F(s-a)$, the inverse Laplace transform will be the original time function $f(t)$ multiplied by $e^{\alpha t}$

Property 4: Translation in the Time Domain (Piecewise)

Also known as the Time Shift Theorem:

• If you delay a function in time, this looks like you’re “damping” or attenuating the function by a factor in the Laplace world

Unit Step Function $H(t)$

$u(t-a)$ is the unit step function that shifts the function $f(t)$ by a time delay $a$

• When we apply the Laplace transform to a time shifted function, the transform incorporates a multiplicative exponential factor
• $\mathcal{L}(t(t-a)) = \frac{e^{-as}}{s}$