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Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 9

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 4 part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | We use the convolution theorem to find the inverse Laplace transform of y s . . f 1 . 7 X 7 X y t 7 sin 2T cos t T dT cos t 02 - Ị sin t T sin 3T t dT cos t 1 4 1 4 cos t T - - cos 3t t cos t 3 J 0 cos 2t cos t - cos 2t - cos t j cos t I cos 2t I cos t 3 3 Alternatively we can find the inverse Laplace transform of y s by first finding its partial fraction expansion. s 3 s 3 s s2 4 s2 1 4s 3 j i 2t 4 cos t 3 y s s 2 1 s 3 s2 4 y t 1 cos 3 Example 31.4.3 Consider the initial value problem y 5y 2y 0 y 0 1 y 0 2. Without taking a Laplace transform we know that since y t 1 2t O t2 the Laplace transform has the behavior y s - s 2 O s-3 as s to. 1494 31.5 Systems of Constant Coefficient Differential Equations The Laplace transform can be used to transform a system of constant coefficient differential equations into a system of algebraic equations. This should not be surprising as a system of differential equations can be written as a single differential equation and vice versa. Example 31.5.1 Consider the set of differential equations yi y2 y2 y3 y3 -y3 - y2 - yi t3 with the initial conditions yi 0 y2 0 y3 0 0. We take the Laplace transform of this system. syi - yi 0 y2 sy2 - y2 0 y3 sy3 - y3 0 -y3 - y2 - y1 si The first two equations can be written as y3 yi 5 s2 y2 T s .