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Solve \(x''+x=f(t), x(0)=0,x'(0)=0\) using Laplace transform. Exercise \(\PageIndex{6.2.12}\) Find the transfer function for \(mx'' + cx'+kx =f(t)\) (assuming the initial conditions are zero).
ANSWERS TO PRACTICE PROBLEMS CHAPTER 6 AND 7 I. Laplace Transform 1. (a) Using the double angle trigonometric identity, the function f t can be rewritten as f t = 1 2 sin 4t . Thus L{f t }= 2 s2 16 (b) Using the half angle trigonometric identity, the function f t can be rewritten as f t = 1 2 1 cos 6t . Thus L{f t }= 1
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Hint 3: Use the sifting property of the delta function, which allows you to pull the function e i!t out of the integral while inserting the value of t at which the delta function has non-zero value.
Class exercises on the Laplace transform, elementary properties and formulas, derivative formulas, solving ordinary differential equation, discontinuous functions, convolution and delta function, and Green’s function.
Section 14-3: Laplace Transform P14.3-1 P14.3-2 P14.3-3 P14.3-4 ss b Section 14-4: Impulse Function and Time Shift Property P14.4-1 P14.4-2 () () ( ) ()() 1 22 11 22 cos Af t AF s ft t Fs s s As Fs s ωω ω = =⇒=+ ∴= + From Table 14.4-1: ft t Fsn s s s n n == ∴= +,!, 1 2 1 but we have n =1 F () ( ) () ( ) sT ()sT ft Aut utT Ae 1e Fs A u ...
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
