tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 10

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 4 part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | We take the Laplace transform of the differential equation. s2y s - sy 0 - y 0 - y s f s s2y s - s - y s f s s f s y s s- -1 By inspection of a table of Laplace transforms we see that L-1 s 2 cosh t H t s2 - 1 L 1 -------- sinh t H t . s2 - 1_ Now we use the convolution theorem. V t L-1 r f r sinh t - r dr s2 - 1 Jo The solution for positive t is y t cosh t Ị f r sinh t - r dr. Clearly the solution is continuous because the integral of a bounded function is continuous. The first derivative of the solution is y t sinh t f t sinh 0 Ị f r cosh t - r dr y t sinh t Ị f r cosh t - r dr We see that the first derivative is also continuous. 1534 Solution We consider the problem dt y T dT e-t y 0 1. We take the Laplace transform of the equation and solve for y. sy - y 0 y -77 s s 1 s s 2 y s 1 s2 1 We expand the right side in partial fractions. 1 1 3s y - 2 s 1 2 s2 1 We use a table of Laplace transforms to do the inversion. y -2e sin t 3cos t Solution We consider the problem L rr Rii q C E0 dt L r RÌ2 q C 0 dt dq dt i1 - Ì2 i1 0 i2 0 0 9 0 0. 2R .