Question

i) Using the convolution theorem, show that h(t) = L?¹(H(s)) = 1/(2w²) (1/w sin wt - t cos wt ), where H(s) = 1/(s²+w²)². ii) Use the method of Laplace transforms to solve the ordinary differential equation d²y/dt² + 3dy/dt + 2y = e??, subject to the initial conditions y = dy/dt = 0 at t = 0. Question 2 Denote the Fourier series of f(x) = { x, -? < x < 0; 0, 0 ? x ? ? } by F(x). Show that F(x) = -?/4 + 2/? ????? cos[(2m+1)x]/(2m+1)² + ????? (-1)??¹/n sin(nx).

          i) Using the convolution theorem, show that h(t) = L?¹(H(s)) = 1/(2w²) (1/w sin wt - t cos wt ), where H(s) = 1/(s²+w²)².

ii) Use the method of Laplace transforms to solve the ordinary differential equation d²y/dt² + 3dy/dt + 2y = e??, subject to the initial conditions y = dy/dt = 0 at t = 0.

Question 2
Denote the Fourier series of f(x) = { x, -? < x < 0; 0, 0 ? x ? ? } by F(x). 
Show that F(x) = -?/4 + 2/? ????? cos[(2m+1)x]/(2m+1)² + ????? (-1)??¹/n sin(nx).

i) Using the convolution theorem, show that h(t) = L?¹(H(s)) = 1/(2w²) (1/w sin wt - t cos wt ), where H(s) = 1/(s²+w²)².

ii) Use the method of Laplace transforms to solve the ordinary differential equation d²y/dt² + 3dy/dt + 2y = e??, subject to the initial conditions y = dy/dt = 0 at t = 0.

Question 2
Denote the Fourier series of f(x) = x, -? < x < 0; 0, 0 ? x ? ? by F(x).
Show that F(x) = -?/4 + 2/? ????? cos[(2m+1)x]/(2m+1)² + ????? (-1)??¹/n sin(nx).

Added by Sandra M.

Mathematical Methods in the Physical Sciences

Mary L. Boas 3rd Edition

Chapter 8

Instant Answer

Solved by Expert Sri K

03/15/2023

Step 1

The given equation is: $$4y'' + 3y' + 2y = e^{-6t}$$ Taking the Laplace transform of both sides, we get: $$4s^2Y(s) - 4sy(0) - 4y'(0) + 3sY(s) - 3y(0) + 2Y(s) = \frac{1}{s+6}$$ Now, we plug in the initial conditions $y(0) = 0$ and $y'(0) = 0$. This simplifies Show more…

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Using the convolution theorem, show that h(t) = L^(-1)(H(s)) where H(s) Use the method of Laplace transforms to solve the ordinary differential equation subject to the initial conditions. Question 2 Denote the Fourier series of f(x) by F(x). Show that F(x)