Laplace transform with boundary conditions

RelativelyGood · Mar 30, 2012

Here's the question:

Use laplace transforms to find X(t), Y(t) and Z(t) given that:

X'+Y'=Y+Z
Y'+Z'=X+Z
X'+Z'=X+Y

subject to the boundary conditions X(0)=2, Y(0)=-3,Z(0)=1.

Now I have learnt the basics of laplace transforms, but have not seen a question in this form before. How do I start the question, could someone for instance show me how to get X(t) and I'll try the rest knowing how to do it? I have other questions I need to do like this, but this looks like the easiest one.

Thanks

_Eduardo_ · Mar 30, 2012

First some cleaning

(1) X'+Y'=Y+Z
(2) Y'+Z'=X+Z
(3) X'+Z'=X+Y

combining (1)-(2)+(3) , (1)+(2)-(3) y (2)+(3)-(1) is:

X' = Y
Y' = Z
Z' = X

Applying Laplace transform

s X(s) - x0 = Y(s)
s Y(s) - y0 = Z(s)
s Z(s) - z0 = X(s)

You could have applied LT from the beginning, only the system would have more terms, but obviously, the same final solution.

Now, you must solve this 3x3 system with unknowns X(s), Y(s), Z(s).
I don't know what methods for solving system you can use, but I will use matrix+software :smile:

[X(s);Y(s);Z(s)] = [s, -1, 0; 0, s, -1; -1, 0, s]^(-1)·[2; -3; 1]

That is:
X(s) = (2·s - 1)/(s^2 + s + 1)
Y(s) = - (3·s + 2)/(s^2 + s + 1)
Z(s) = (s + 3)/(s^2 + s + 1)

Now, you must calculate the inverse Laplace transform.
Again, I don't know what methods you can use, but I will use WolframAlpha :smile:

X(t) = InverseLaplaceTransform[(2·s - 1)/(s^2 + s + 1)]
Y(t) = InverseLaplaceTransform[- (3·s + 2)/(s^2 + s + 1)]
Z(t) = InverseLaplaceTransform[(s + 3)/(s^2 + s + 1)]

See the results on the links

RelativelyGood · Mar 30, 2012

_Eduardo_ said:
First some cleaning

(1) X'+Y'=Y+Z
(2) Y'+Z'=X+Z
(3) X'+Z'=X+Y

combining (1)-(2)+(3) , (1)+(2)-(3) y (2)+(3)-(1) is:

X' = Y
Y' = Z
Z' = X

Applying Laplace transform

s X(s) - x0 = Y(s)
s Y(s) - y0 = Z(s)
s Z(s) - z0 = X(s)

You could have applied LT from the beginning, only the system would have more terms, but obviously, the same final solution.

Now, you must solve this 3x3 system with unknowns X(s), Y(s), Z(s).
I don't know what methods for solving system you can use, but I will use matrix+software :smile:

[X(s);Y(s);Z(s)] = [s, -1, 0; 0, s, -1; -1, 0, s]^(-1)·[2; -3; 1]

That is:
X(s) = (2·s - 1)/(s^2 + s + 1)
Y(s) = - (3·s + 2)/(s^2 + s + 1)
Z(s) = (s + 3)/(s^2 + s + 1)

Now, you must calculate the inverse Laplace transform.
Again, I don't know what methods you can use, but I will use WolframAlpha :smile:

X(t) = InverseLaplaceTransform[(2·s - 1)/(s^2 + s + 1)]
Y(t) = InverseLaplaceTransform[- (3·s + 2)/(s^2 + s + 1)]
Z(t) = InverseLaplaceTransform[(s + 3)/(s^2 + s + 1)]

See the results on the links

Thank you. I posted this in a hope to get an answer as thorough and clear, so that I can now complete harder questions, knowing the method. I have managed to do one by myself, by hand, so thanks!

Laplace transform with boundary conditions

RelativelyGood

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_Eduardo_

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