Convert complex ODE to matrix form

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!

 

[haruspex]

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?

 

[eys_physics]

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.