Convert complex ODE to matrix form

In summary, the conversation discusses converting a second order differential equation into a system of two differential equations by introducing a new variable. However, the proposed matrix form does not make sense for a single equation and the choice of coefficients for the second row is unclear. It is also mentioned that the equations only hold under a special condition, which is not implied by the given differential equation.
  • #1
SeM
Hi, I have the following complex ODE:

aY'' + ibY' = 0

and thought that it could be written as:

[a, ib; -1, 1]

Then the determinant of this matrix would give the form

a + ib = 0

Is this correct and logically sound?

Thanks!
 
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  • #2
As far as I am aware, writing in matrix form is for when you have a system of differential equations in correspondingly multiple variables. I do not see how it can be applied to a single equation. On what would you base the choice of coefficients for the second row?
 
  • #3
SeM said:
Hi, I have the following complex ODE:

aY'' + ibY' = 0

and thought that it could be written as:

[a, ib; -1, 1]

Then the determinant of this matrix would give the form

a + ib = 0

Is this correct and logically sound?

Thanks!

Hey, SeM,

I don't understand how you arrive at the second row in the matrix. Maybe, what you want to convert the second order differential equation into a system of two differential equations. This can be done by introducing
$$W(x)=Y'(x),$$
leading to the equations
$$aW'(x)+ibW(x)=0, \quad Y'(x)=W(x)$$

Indeed, both of your equations, i.e.
$$aY''(x)+ibY'(x)=0, Y''(x)-Y'(x)=0$$
if
$$a+ib=0,$$ but this is only a special case and it doesn't follow from your stated differential equation.
 

1. How do I convert a complex ODE to matrix form?

To convert a complex ODE to matrix form, you can start by writing the ODE in its general form, with all the terms on one side and the dependent variable on the other side. Then, you can replace the dependent variable with a vector of variables, and the derivatives with a matrix of partial derivatives. This will result in a system of linear equations in matrix form.

2. What is the benefit of converting a complex ODE to matrix form?

Converting a complex ODE to matrix form can make it easier to solve and analyze. It allows for the use of linear algebra techniques, which can simplify the process and provide a better understanding of the behavior of the system.

3. How can I determine the dimension of the matrix for a converted ODE?

The dimension of the matrix for a converted ODE will depend on the number of variables and derivatives in the original ODE. For example, if the original ODE has two variables and two derivatives, the resulting matrix will be a 2x2 matrix.

4. Can all complex ODEs be converted to matrix form?

Not all complex ODEs can be converted to matrix form. Some ODEs may have nonlinear terms or special functions that cannot be represented in a matrix. It is important to carefully consider the structure of the ODE before attempting to convert it to matrix form.

5. Are there any limitations to solving a converted ODE in matrix form?

Solving a converted ODE in matrix form may have limitations depending on the size and complexity of the matrix. Large matrices can be computationally expensive to solve, and some systems may have numerical stability issues that can affect the accuracy of the solution. It is important to consider these limitations when using this method of solving ODEs.

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