Laplace tranforms with boundary conditions

[wtmoore]

Homework Statement

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 learned 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

Homework Equations

The Attempt at a Solution

 

[LCKurtz]

Take the transform of each equation so instead of 3 equations in X(t), Y(t), and Z(t) you have 3 equations in their transforms x(s), y(s), and z(s). Then solve the 3 equations for those three transforms.

 

[wtmoore]

I am still unsure.

Take L() to be notation for laplace.

Top line,

L(X')=sx(s)-X(0)
L(Y')=sy(s)-Y(0)
L(Y)=y(s)
L(Z)=z(s)

How do I solve from here?

sx(s)-X(0)+sy(s)-Y(0)=y(s)+z(s)

 

[LCKurtz]

I am still unsure.

Take L() to be notation for laplace.

Top line,

L(X')=sx(s)-X(0)
L(Y')=sy(s)-Y(0)
L(Y)=y(s)
L(Z)=z(s)

How do I solve from here?

sx(s)-X(0)+sy(s)-Y(0)=y(s)+z(s)

Sorry for the delay in getting back. Google has started intercepting Forum posts as spam and I didn't know it. What you need to do is take the LaPlace transform of both sides of all three equations, using formulas like you have listed above. You will get 3 equations in the three unknowns x(s), y(s), and z(s). And you know X(0) = 2 and Y(0)=-3 so use that. Plug the equations above into the first equation. Then do likewise with the other two.