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Pierre Simon Laplace

It’s not “This is the place” but this is Laplace. Pierre Simon Laplace. L [f( t )]. Initial value problem. Algebraic equation for X ( s ). Take Laplace transform. obstacle. X ( s ) = F ( s ). Take inverse Laplace transform. Solution x ( t ). L -1 [ F ( s )].

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Pierre Simon Laplace

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  1. It’s not “This is the place” but this is Laplace Pierre Simon Laplace

  2. L[f(t)] Initial value problem Algebraic equation for X(s) Take Laplace transform obstacle X(s) = F(s) Take inverse Laplace transform Solution x(t) L-1[F(s)] Steps to Solving in IVP using the Laplace Transform Solve algebraically for X(s)

  3. Let f be a function defined on the interval [0, ). The Laplace transform of f is the function F(s) defined by provided that the improper integral converges. The Laplace transform of f is denoted by L[f] Determine the Laplace transform of the following functions:

  4. is a linear transform The Laplace transform

  5. A function f is said to be of exponential orderif there exist constants M and  such that for all t > T. So if we can bound f(t) with an exponential function, the Laplace transform exists. Show that the given function is of exponential order:

  6. A function f is called piecewise continuouson the interval [a, b] is we divide [a, b] into a finite number of subintervals in such a manner that • f is continuous on each subinterval, and • 2. f approaches a finite limit as the endpoints of each subinterval are approached from within.

  7. Find the Laplace transform L[f ] when

  8. Sketch the given function and determine its Laplace transform:

  9. The linear transformation defined by Is called the inverse Laplace transform. Find L-1[F](t) if:

  10. The First Shifting Theorem

  11. Solve the IVP: Suppose that f is of exponential order on [0, ) and that f' exists and is piecewise continuous on [0, ). Then L[ f' ] exists and is given by • Take the Laplace transform of the given DE, and substitute in the given initial conditions. • Solve the resulting equation algebraically for Y(s). • Take the inverse Laplace transform of the resulting equation to determine the solution y(t) of the given IVP.

  12. Solve the IVP: What if the DE is second-order??? What about higher order???

  13. Solve the IVP: Solve the IVP: Solve the IVP: Solve the IVP:

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