M427K Handout: Applications of the Laplace Transform: StepFunctions and Discontinuous Forcing FunctionsSalman ButtMarch 31, 2006Notes1. I gave this puzzle before break: Two men are standing in front of a house. One man says tothe other, “I want you to guess the ages of my 3 daughters. Here are two clues: Their agesadd up to the sum of the number of this house. The product of their ages is 36.” After a fewminutes, the second man turns to the first and says, “I don’t have e nough information andneed another clue.” The first man says, “My eldest daughter has red hair.” What are theages of the first man’s daughters?The first thing you want to do is write out all possible factorizations of 36 into 3 numbers:1 · 1 · 36 1 · 2 · 18 1 · 3 · 12 1 · 4 · 9 1 · 6 · 6 2 · 1 · 18 2 · 2 · 9 2 · 3 · 6 3 · 1 · 12 3 · 2 · 63 · 3 · 4 3 · 4 · 3 3 · 6 · 2 4 · 1 · 9 4 · 3 · 3 4 · 9 · 1 6 · 1 · 6 6 · 2 · 3 6 · 3 · 2 6 · 6 · 19 · 1 · 4 9 · 2 · 2 9 · 4 · 1 12 · 3 · 1 18 · 2 · 1 36 · 1 · 1Applications of the Laplace Transform: Step FunctionsWe will assume all of our functions below are piecewise continuous and of exponential order (i.e.their Laplace transforms exist). We define the unit step functions (or Heaviside functions) asuc(t) =where c ≥ 0. Here are the graphs of uc(t) and 1 − uc(t):The functions are helpful in dealing with functions that have jump discontinuities. Their Laplacetransform is easily computed:L {uc(t)} (s) =1We will need to translate functions by some positive amount c, which is done using these stepfunctions. So given a function f(t) defined for t ≥ 0, we can translate this function by c to get anew function g(t) = uc(t)f(t − c):It should not surprise you that the Laplace transforms of f(t) and uc(t)f(t − c) are related:Theorem 1. If the Laplace transform of f exists for s greater than some con stant a ≥ 0 and if ca positive constant, then for s > a we haveL {uc(t)f(t − c)} (s) =Observe that if we use the above relation with f(t) = 1, we findAs an example, let’s compute the Laplace transform of the following function:f(t) =sin t, 0 ≤ t < π/4sin t + cos(t − π/4), t ≥ π/4We also have the following result which proves very useful:Theorem 2. If the Laplace transform of f exists for s greater than some con stant a ≥ 0 and if cis a positive constant, then for s > a + c, we haveLectf(t)(s) =As an example of this theorem, consider the inverse Laplace transform of G(s) = 1/(s2−4s+5):2Applications of the Laplace Transform: Discontinuous Forcin g FunctionsWe will now use the work done above to solve a second-order differential equation which hasa discontinuous forcing term. The equation has a physical meaning (modeling the charge on thecapacitor in a simple circuit with unit voltage pulse for 5 ≤ t < 20, or as the response of a dampedoscillator subject to the given force g(t) be low), so its interest is beyond purely mathematical: Solvethe differential equation2y00+ y0+ 2y = g(t), y(0) = 0, y0(0) = 0, with g(t) = u5(t) − u20(t) =1, 5 ≤ t < 200, 0 ≤ t < 5, t ≥