Section 8.4: Solving IVPs using Laplace Transforms

Derivatives and Antiderivatives Remember how in Calculus 1, there was an explicit definition of the derivative of a function involving limits. This was something that could be evaluated using algebra and simplifying and then letting h go to zero. Contrast this to the way the antiderivative was defined. We say that F (t) is the antiderivative of f (t) if F 0 (t) = f (t). Thus, instead of having a formula for computing the antiderivative, we had to rely on our knowledge of derivatives and look at the form that f (t) was in to be able to say what function would have a derivative that is equal to f (t). It is the same way with Laplace Transforms and Inverse Laplace Transforms. . . Definition Let f (t) be a function whose Laplace transform, F (s) exists for s > s0 . We say that f (t) is the inverse Laplace Transform for F (s) for t > 0 and write f (t) = L−1 {F (s)}(t). Thus, F (s) = L{f (t)}(s) f (t) = L−1 {F (s)}(t) are equivalent statements. Brief Table of Inverse Laplace Transforms and Properties F (s) f (t) 1 (s−α)n 1 αt n−1 (n−1)! e t 1 (s−α2 )+β 2 s−α (s−α)2 +β 2 1 αt βe e αt cos(βt) F (s − a) e at L−1 {F (s)}(t) e −bs F (s) u(t − b)L−1 {F (s)}(t − b) e −cs δ(t − c) sin(βt) Example Find the inverse Laplace Transform of the following functions. 1 I s−3 I I I s−2 s 2 +9 s−2 (s−2)2 +9 4 (s−3)3 I e −2s s e −4s (s−4) (s−4)2 +9 I e −2s 6 I Solving IVPs using Laplace Transforms To solve a linear IVP: 1. Take the Laplace Transform of the ODE. This means the Laplace Transform of the LHS is equal to the Laplace Transform of the RHS. 2. Plug in your initial values. 3. Solve for Y (s). 4. Then, y (t) = L−1 {Y (s)}(t) is the solution to the ODE. Example Solve the IVP using Laplace Transforms. y 00 + 2y 0 + y = e −t , y (0) = 1, y 0 (0) = 0 Example Solve the IVP using Laplace Transforms. y 00 + 4y 0 + 8y = δ(t − 3), y (0) = 1, y 0 (0) = −2 Example Solve the IVP using Laplace Transforms. ty 00 − ty 0 + y = 2, y (0) = 2, y 0 (0) = −4 Example A fluid tank contains 1000 Liters of a brine solution (salt and water). Let x(t) be the amount of salt in the tank at time t in kilograms, and suppose x(0) = 0 kg. A brine solution flows into the tank at a rate of 6L/min with a salt concentration of f (t), and water exits the tank at the same rate. Assume that the solution is well mixed. The amount of salt in the brine solution is modeled by the following IVP. 3 dx + x(t) = 6f (t), x(0) = 0 dt 500 If the inflow function, f (t), is given by ( .5kg /L, if 0 ≤ t ≤ 10 f (t) = .1kg /L, if t ≥ 10 Find x(t).