🗝️ Key Takeaways ...

• Step functions are mathematical ON/OFF switches that let you control when a function is active.
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• The unit step function \(u(t)\) is given by

  \begin{equation*} u(t) = \left\{ \begin{array}{ll} 1, \amp t \ge 0 \\ 0, \amp t \lt 0 \end{array} \right. \end{equation*}

  jumps from \(0\) to \(1\) at \(t=0\text{.}\) We say it activates at \(t=0\text{.}\)

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• To active a function at \(t=0\text{,}\) multiply it by \(u(t)\text{.}\)
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• You can change the activation point by shifting \(u(t)\) horizontally:

  \begin{equation*} u_c(t) = u(t-c) = \left\{ \begin{array}{ll} 0, \amp t \lt c \\ 1, \amp t \ge c \end{array} \right. \end{equation*}

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• Multiplying by \(u_c(t)\) activates a function at \(t = c\text{:}\)

  \begin{equation*} f(t)\,u_c(t) = \begin{cases}0,& t \lt c\\ f(t),& t \ge c\text{.}\end{cases} \end{equation*}

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• By default, \(u_c(t)\) is an OFF-ON switch, but you can turn it into an ON-OFF switch with:

  \begin{equation*} 1-u_c(t) = \left\{ \begin{array}{ll} 1, \amp t \lt c \\ 0, \amp t \ge c \end{array} \right. \end{equation*}

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• So multiplying by \(1-u_c(t)\) deactivates an active function at \(t = c\text{:}\)

  \begin{equation*} f(t)\,(1 - u_c(t)) = \begin{cases}f(t),& t \lt c\\ 0,& t \ge c\text{.}\end{cases} \end{equation*}

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• To activate a function only on a window, multiply:

  \begin{equation*} f(t)\,(u_c(t) - u_d(t)) = \begin{cases}f(t),& c \le t \lt d\\ 0,& \text{otherwise}\text{.}\end{cases} \end{equation*}

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• When a differential equation has a piecewise forcing term, start the Laplace method process by rewriting it using step functions.
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• Combine like step terms to minimize how many Laplace transforms you have to compute.
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• After rewriting the forcing function, apply the same three Laplace steps you already know: forward transform, solve for \(Y(s)\text{,}\) and invert.
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