# Laplace tranforms with boundary conditions

• wtmoore
In summary: You have three equations in three unknowns. Solve for the unknowns using any method you like.In summary, to find X(t), Y(t), and Z(t) using Laplace transforms, one must first take the transform of each equation and solve for the transforms of the unknown functions. Then, plug the transformed equations into the original equations and use the given boundary conditions to solve for the unknowns.
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

## The Attempt at a Solution

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.

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)

wtmoore said:
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.

## 1. What is a Laplace transform and how is it used in boundary value problems?

A Laplace transform is a mathematical tool used to convert a function of time into a function of complex frequency. It is commonly used in solving boundary value problems because it can transform differential equations into algebraic equations, making them easier to solve.

## 2. What are boundary conditions and why are they important in Laplace transforms?

Boundary conditions are constraints or requirements that must be satisfied at the boundaries of a system or problem. In Laplace transforms, they are important because they provide additional information that can help determine the solution to the problem.

## 3. Can Laplace transforms be used for both initial value problems and boundary value problems?

Yes, Laplace transforms can be used for both initial value problems (IVPs) and boundary value problems (BVPs). However, they are more commonly used for BVPs because they are better suited for solving problems with boundary conditions.

## 4. How do you apply boundary conditions when using Laplace transforms?

When using Laplace transforms to solve a BVP, the boundary conditions are applied by substituting them into the transformed equation. This will result in a system of equations that can be solved to find the constants in the solution.

## 5. Are there any limitations to using Laplace transforms with boundary conditions?

One limitation of using Laplace transforms with boundary conditions is that it can only be used for linear differential equations. Additionally, the boundary conditions must be well-defined and solvable in order for the transformed equation to have a solution.

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