- #1

fufufu

- 17

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## Homework Statement

use laplace transforms to solve system of two equations

x' = x -4y + e^4t

y' = -x + y -4e^4t

x(0) = 0

y(0) =1

## Homework Equations

i am uncertain about the correct order to solve these problems.. am I taking inverse L.T. at right time and place in order for the prob to check out? thanks for any help

## The Attempt at a Solution

sX(s) - 0 - X(s) = -4Y(s) + 1/s-4

X(s) = -4Y(s)/s-1 + 1/(s-4)(s-1)

sY(s) -1 -Y(s) - 4Y(s)/s-1) = -1/(s-4)(s-1)

Y(s) = 1/(s-1)(s-3)(s-4) + 1/s-3

1 = A(s-1)(s-3) + B(s-4)(s-3) + C(s-4)(s-1)

after letting s = 1, 3 and 4 i get

A=1/3, B=1/6 and C=-1/2

Y(s) = 1/3(1/s-4) + 1/6(1/s-1) -1/2(1/s-3)

y(t) = (1/3)e^4t + (1/6)e^t - (1/2)e^3t

y(t) = e^t

X(s) = -4Y(s)(1/s-1) + 1/(s-4)(s-1)

after using partial fractions on rightmost term on RHA and inserting y(t) for other term i get

X(s) = -4e^t(1/s-1) + 1/(s-4)(s-1)

x(t) = -4e^t(e^t) + (1/3)(1/s-4) - (1/3)(1/s-1)

x(t) = -4e^t(e^t) + (1/3)e64t - (1/3)e^t

final answer:

x(t) = -4e^2t + (1/3)e^4t - (1/3)e^t

y(t) = e^t