3. (25 Points) Find f(t). f(0) + f(t - 1)f(t)dt = t. Hint: The second term...
Use the Laplace transform to solve initial value problems 5. *" + 4x = f(x); x(t) = 35. f(t – 1) sin 27 dt, x(0) = x'(0) = 0 (use a convolution theorem).
(6). The quantities x(t) and y(t) satisfy the simultaneous equations dt dt dx dt where x(0)-y(0)-ay (0)-0, and ax (0)-λ. Here n, μ, and λ are all positive real numbers. This problem involves Laplace transforms, has three parts, and is continued on the next page. You must use Laplace transforms where instructed to receive credit for your solution (a). Define the Laplace Transforms X(s) -|e"x(t)dt and Y(s) -e-"y(t)dt Laplace Transform the differential equations for x(t) and y(t) above, and incorporate...
4. (10 points) Use the method of convolution to find (a) (5 points) L-'{[82 +1)62+1)} (b) (5 points) L{f(t)}, where f(t) = S.sin(27) cos(t – )dt 5. (20 points) Using the method of Laplace transform, solve y" – y' – 2y = 0, y(0) = 0, y'(0) = 1,
Useful Formula: Fourier Transform: F[f(t)] = F(w) sof(t)e-jw dt Inverse Fourier Transform: F-1[F(w)] = f (t) = 24., F(w)ejwidw Time Transformation property of Fourier Transform: f(at – to). FC)e=itoch Laplace Transform: L[f(t)] = F(s) = $© f(t)e-st dt Shifting property: L[f(t – to)u(t – to)] = e-toSF(s) e [tuce) = 1 and c [u(e) = ) Using the convolution property of Fourier Transform to find the following convolution: sinc(t) * sinc (4t) [Hint: sinc(t) or rect(w/2)] TC .
Solve the following IVPs using Laplace Transform: 1) dy dt 3y(t) = e4t; y(0) = 0
3) Let F(x) = {* In In(1+t) dt. t (a) Find the Maclaurin series for F: (b) Use the series in part (a) to evaluate F(-1) exactly and use the result to state its interval of convergence. (c) Approximate F(1) to three decimals. (Hint: Look for an alternating series. )
2. Using Laplace transform, solve the system of differential equations d.x: dy dt where x(0)1 2. Using Laplace transform, solve the system of differential equations d.x: dy dt where x(0)1
dx Determine x= f(t) for (t? +4t) 4x + 4,t> 0; f(1) = 3. dt For (1? + 4t) dx dt = 4x +4, x= f(t) =
Find the Laplace transform of f(t)=∫ 0 t τsin(2τ) dτ F(s)= Find the Laplace transform of f(t) = Tsin(27) dt F() =
1. (2 points) Using the definition, find the Laplace Transform of the function: e21, 0<t<3 f(t) = 3<t