How do I solve a system of ODE's using Laplace transform?

In summary, the conversation is about solving a system of ODE's by Laplace transforming with initial conditions. The Laplace transformed variables are used to find the inverse Laplace transform, but there is an error in the equations that is pointed out and corrected.
  • #1
Niles
1,866
0

Homework Statement


Hi

I am trying to solve the following system of ODE's by Laplace transforming:
[tex]
x' = 1 + 21y - 6x \\
y' = 6x-53y
[/tex]
with the initial conditions x(0)=y(0)=0. Laplace transforming gives me (X and Y denote the Laplace transformed variables)
[tex]
sX = 1 + 21y-6x \\
sY = 6x-53y
[/tex]
From these I find
[tex]
X(s) = \frac{1}{6+s-126/(s+53)}
[/tex]
The inverse Laplace transform is (I have checked this with Mathematica)
[tex]
x(t) = \frac{e^{-\frac{1}{2} \left(59+\sqrt{2713}\right) t} \left(-47+\sqrt{2713}+\left(47+\sqrt{2713}\right) e^{\sqrt{2713} t}\right)}{2 \sqrt{2713}}
[/tex]
When I take t=0, then I get x(0)=1, not x(0)=0. I not quite sure where I have gone wrong, I have double-checked everything by doing it numerically too.

Is there something that I have forgotten to do?Niles.
 
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  • #2
Niles said:

Homework Statement


Hi

I am trying to solve the following system of ODE's by Laplace transforming:
[tex]
x' = 1 + 21y - 6x \\
y' = 6x-53y
[/tex]
with the initial conditions x(0)=y(0)=0. Laplace transforming gives me (X and Y denote the Laplace transformed variables)
[tex]
sX = 1 + 21y-6x \\
sY = 6x-53y
[/tex]
Shouldn't these be$$
sX =\frac 1 s +21Y-6X$$ $$
sY = 6X-53Y$$
 
  • #3
You are right, thanks for that! I don't know why I thought it would just be constant.Niles.
 

1. What is an ODE?

An ODE (ordinary differential equation) is a mathematical equation that describes how a quantity changes over time, based on its current value and the rate at which it is changing. It is commonly used to model physical, biological, and economic systems.

2. What is a Laplace transform?

A Laplace transform is a mathematical operation that converts a function of time into a function of complex frequency. It is often used to solve ODEs by transforming them into algebraic equations, which are easier to solve.

3. How is a Laplace transform used to solve ODEs?

By taking the Laplace transform of both sides of an ODE, the equation is transformed into an algebraic equation that can be solved for the desired function. After solving for the transformed function, the inverse Laplace transform is then used to find the solution to the original ODE.

4. What are some advantages of using Laplace transforms to solve ODEs?

One advantage is that it can simplify the process of solving ODEs, particularly for higher-order equations. It also allows for the use of algebraic techniques, which can be more straightforward than traditional methods such as separation of variables or power series solutions.

5. Are there any limitations to using Laplace transforms for solving ODEs?

Yes, one limitation is that the function must have a Laplace transform in order for this method to be applicable. Additionally, it may not be as intuitive as other methods for solving ODEs, and some knowledge of complex analysis may be required for more advanced problems.

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