Using laplace to solve 2 equationsplease check workthanks

[fufufu]

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

 

[mighty2000]

Thank you for your post. Your approach to solving the system of equations using Laplace transforms seems correct. However, there are a few minor errors in your calculations that I would like to point out.

Firstly, in the equation for Y(s), the term -4Y(s)/(s-1) should be -4Y(s)/(s+1). This is because the transform of e^4t is 1/(s-4), so when you substitute s=4, you should get 1/(4-4) = 1, not 0.

Secondly, in the calculation for the constants A, B, and C, you have used the wrong values for s. When substituting s=1, 3, and 4, you should get 1=A(1-1)(1-3) + B(4-4)(4-3) + C(4-4)(4-1), which simplifies to 1=0+0+C(0), so C=1. Similarly, when substituting s=1 and 3, you should get 1=A(1-1)(1-3) + B(3-4)(3-3) + C(3-4)(3-1), which simplifies to 1=0+B(0)+0, so B=1. Finally, when substituting s=1, you should get 1=A(1-1)(1-3) + B(1-4)(1-3) + C(1-4)(1-1), which simplifies to 1=0+0+C(0), so C=1. Therefore, the correct values for the constants are A=1/3, B=1/3, and C=1.

Finally, in your final answer for x(t), the term -4e^t(e^t) should be -4e^4t. This is because you have already substituted s=4 in the equation for X(s), so the term e^4t becomes 1/(4-4) = 1, not e^4t.

With these corrections, your final answer for x(t) and y(t) becomes:

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