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Unit 9: The Laplace Transform

Turning a differential equation into algebra: transforms, inverses, initial value problems in one pass, step and impulse forcing, and convolution.

14 topics · 46 questions

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6.1 What the Laplace Transform Does

  1. How it worksHow does the Laplace transform trade a function of t for a new function of s built from an integral?
  2. Common mistakeWhy isn't the Laplace transform just a table lookup with nothing actually happening underneath it?
  3. Going furtherWhat happens to a Laplace transform's usefulness once you also need it to handle negative time?
  4. Worked exampleFind the Laplace transform of f(t) = 5 using the definition's integral directly.
Your turn1 of 2
Find the Laplace transform of f(t) = 7, evaluated at s = 3.

6.1 The Definition of the Laplace Transform

  1. How it worksHow does the improper integral of e−st f(t) from 0 to infinity define the new function F(s)?
  2. Worked exampleFind the Laplace transform of f(t) = e3t directly from the definition.
  3. Going furtherWhat happens to the Laplace transform's integral when f(t) grows faster than any exponential?
Your turn1 of 2
Find the Laplace transform F(s) of f(t) = e2t directly from the definition (valid for s > 2).

6.1 Transforms of Common Functions

  1. How it worksHow does integrating e−st times tn by parts repeatedly produce the pattern n!/sn+1 for the transform of tn?
  2. Worked exampleFind the Laplace transform of f(t) = t2 using the pattern for powers of t.
  3. Worked exampleFind the Laplace transform of f(t) = sin(4t).
Your turn1 of 2
Find the Laplace transform F(s) of f(t) = 4t2, using the pattern n!/sn+1.

6.1 Linearity of the Laplace Transform

  1. How it worksHow does the integral defining the Laplace transform inherit linearity straight from the linearity of integration?
  2. Worked exampleFind the Laplace transform of f(t) = 3t2 - 5cos(2t) + 4e−t.
  3. Common mistakeWhy isn't the Laplace transform of a product of two functions just the product of their two transforms?
Your turn1 of 2
Find the Laplace transform F(s) of f(t) = 4t2 - 4cos(5t) + 2e−t.

6.2 The Inverse Laplace Transform

  1. How it worksHow do you recognize a transform F(s) as matching a known pattern to read off its inverse?
  2. Worked exampleFind the inverse Laplace transform of F(s) = 3/(s2 + 9).
  3. Going furtherWhat happens when a transform F(s) doesn't obviously match any of the standard patterns in a table?
  4. Worked exampleFind the inverse Laplace transform of a sum of two simple fractions, F(s) = 4/(s - 2) + 1/s.
Your turn1 of 2
Find the inverse Laplace transform f(t) of F(s) = 6/(s2 + 42).

6.2 Partial Fractions and Inverse Transforms

  1. How it worksHow do you decompose F(s) = (s+3)/((s+1)(s+2)) into a sum of simpler terms with unknown constants?
  2. Worked exampleFind the inverse Laplace transform of F(s) = (2s + 3)/(s2 - s - 6) using partial fractions.
  3. Worked exampleFind the inverse Laplace transform of F(s) = 5/(s2 + 4s + 13) by completing the square first.
Your turn1 of 2
Find the inverse Laplace transform f(t) of F(s) = (s + 3)/((s + 2)(s + 6)) using partial fractions.

6.2 The Transform of a Derivative

  1. How it worksHow does integration by parts turn the Laplace transform of y' into sY(s) minus y(0)?
  2. Worked exampleFind the Laplace transform of y'' in terms of Y(s), y(0), and y'(0).
  3. Common mistakeWhy doesn't the Laplace transform of a derivative erase the initial conditions the way plain differentiation would?
Your turn1 of 2
Given that L{y'} = sY(s) - y(0), find L{y''} = s2 Y(s) - s y(0) - y'(0), evaluated at Y(s) = 4, y(0) = 3, y'(0) = 4, s = 3.

6.2 Solving an Initial Value Problem with the Laplace Transform

  1. How it worksHow do you transform both sides of a differential equation, solve for Y(s), and invert to get y(t)?
  2. Worked exampleSolve y'' + 3y' + 2y = 0, y(0) = 1, y'(0) = 0 using the Laplace transform.
  3. Worked exampleSolve y'' + 4y = sin(t), y(0) = 0, y'(0) = 0 using the Laplace transform.
Your turn1 of 2
Solve y'' + (2+4)y' + 2·4y = 0, y(0) = 3, y'(0) = -1 using the Laplace transform. Find y(t).

6.3 The Unit Step Function

  1. How it worksHow does uc(t) act like an on-off switch that turns a piece of a function on starting at t = c?
  2. Worked exampleFind a unit step function expression for f(t) that equals 3 on 0 ≤ t < 2 and t on t ≥ 2.
  3. Going furtherWhat happens to a function's Laplace transform once it's multiplied by a unit step function that delays it?
  4. Going furtherWhat happens to a unit step function uc(t)'s own Laplace transform as c grows larger and larger?
Your turn1 of 2
A student is asked to write a unit-step-function expression for f(t) that equals 5 on 0 ≤ t < 4 and equals t on t ≥ 4, and proposes f(t) = 5 + t·u4(t). What is wrong with this proposal?

6.3 Transforms of Shifted Functions

  1. How it worksHow does the second shifting theorem turn the transform of uc(t) f(t-c) into e−cs F(s)?
  2. Worked exampleFind the Laplace transform of u3(t)(t - 3)2.
  3. Common mistakeWhy doesn't shifting uc(t) f(t) let you just plug t - c into f's ordinary Laplace transform without adjusting?
Your turn1 of 2
Find the Laplace transform F(s) of u4(t)(t-c)2.

6.5 The Dirac Delta Function and Impulse Forcing

  1. How it worksHow does the delta function act as the limit of an ever-narrower, ever-taller pulse with area one?
  2. Worked exampleFind the Laplace transform of delta(t - 3).
  3. Going furtherWhat happens to a system's response the instant an impulse hits it, compared to just before the impulse arrives?
  4. Worked exampleFind the Laplace transform of 3 delta(t - 2) + delta(t).
Your turn1 of 2
Find the Laplace transform F(s) of delta(t - 4).

6.5 Solving an Equation with an Impulse Force

  1. Worked exampleSolve y'' + 4y = delta(t - 1), y(0) = 0, y'(0) = 0 using the Laplace transform.
  2. Worked exampleSolve y'' + 2y' + 5y = delta(t), y(0) = 0, y'(0) = 0 using the Laplace transform.
  3. Ties togetherHow does an impulse response here connect to the underdamped free vibration you'd get with no forcing at all?
Your turn1 of 2
Solve y'' + 42 y = delta(t - 0.23), y(0) = 0, y'(0) = 0 using the Laplace transform. Find y(t) for t > c.

6.6 The Convolution Integral

  1. How it worksHow does the convolution integral (f·g)(t) slide one function backward across the other and integrate the overlap?
  2. Worked exampleFind the convolution of f(t) = t and g(t) = sin(t).
  3. Common mistakeWhy isn't the convolution of f and g the same as the ordinary product f(t) times g(t)?
Your turn1 of 2
Find the convolution (f·g)(t) of f(t) = 4t and g(t) = sin(t).

6.6 Using Convolution to Solve a Differential Equation

  1. How it worksHow does writing Y(s) as a product of two known transforms let you invert it using convolution instead of partial fractions?
  2. Worked exampleSolve y'' + y = f(t), y(0) = 0, y'(0) = 0, expressing the answer as a convolution integral.
  3. Going furtherWhat happens to the convolution solution once the forcing function f(t) is only known from measured data, not a formula?
Your turn1 of 2
For y'' + 42 y = 6, y(0)=0, y'(0)=0, the convolution solution is y(t) = (k/w2)(1-cos(wt)). Find y(1.42).