Αποτελέσματα Αναζήτησης
PRACTICE PROBLEMS CHAPTER 6 AND 7 I. Laplace Transform 1. Find the Laplace transform of the following functions. (a) f t =sin 2t cos 2t (b) f t =cos2 3t (c) f t =te2tsin 3t (d) f t = t 3 u7 t (e) f t =t2u 3 t (f) f t ={1, if 0≤t 2, t2−4t 4, if t≥2 (g) f t ={t, if 0≤t 3,
Let Y(s) = L[y(t)] be the Laplace transform of the solution. Applying Lto the equation, we obtain the transformed equation L[0] = L[y0] L [y] = sY y(0) Y: Since L[0] = 0; we get 0 = (s 1)Y y(0); which is trivial to solve! The transformed solution to the ODE is then Y(s) = y(0) s 1:
1. Use the rules and formulas to nd the Laplace transform of e t(t2 + 1): 2. Let f(t) = e t cos(3t): (a) From the rules and tables, what is F(s) = L[f(t)]? (b) Compute the derivative f0(t) and its Laplace transform. Verify the t-derivative rule in this case. 3. Use the Laplace transform to nd the unit impulse response and the unit step response
22 CHAPTER 1. LAPLACE TRANSFORM SOLUTIONS Full Solution: The Fourier transform of the time-domain function f(t) is given by Eq. 1.7 as F(!) = ∫1 1 f(t)e i!tdt: Inserting the Dirac delta function (t) into this equation for f(t) gives F(!) = ∫1 1 (t)e i!tdt: This integral can be evaluated by using the sifting property of the
Use Properties and Basic Transforms. Find Laplace Transform. sin(5 t + 2) t. 2 et. e −. sin2 t. t sin t.
this lecture I will explain how to use the Laplace transform to solve an ODE with constant coefficients. The main tool we will need is the following property from the last lecture: 5 Differentiation. Let {f(t) = F (s). Then. } {f′(t)} = sF (s) f(0), −. {f′′(t)} = s2F (s) sf(0) f′(0).
Laplace Transform Practice Problems (Answers on the last page) (A) Continuous Examples (no step functions): Compute the Laplace transform of the given function.
