Repeated complex conjugate roots for Cauchy-Euler

In summary, the general equation for repeated complex conjugate roots in a 4th order Cauchy Euler equation is X^alpha [C1 cos(beta lnx) + C2 sin(beta lnx)^2]. However, the last term may be incorrect and should possibly be C2 sin [beta (lnx)] lnx. It is important to note that repeated roots do not necessarily mean that only C1 and C2 are present, as explained by Maslanka on page 3. This insight was helpful in identifying and correcting an error in the auxiliary equation.
  • #1
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Looking for the general equation for repeated complex conjugate roots in a 4th order Cauchy Euler equation.

This is incorrect, but I think it is close:
X^alpha [C1 cos(beta lnx) + C2 sin(beta lnx)^2]

I think that last term is a little off. Maybe C2 sin [beta (lnx)] lnx ?
 
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  • #2
Repeated roots doesn't mean you only have a c1 and a c2. See Maslanka top of p 3.
 
  • #3
That is the Rosetta stone I needed. Thanks. My auxiliary equation is wrong which is throwing everything else off.
 
  • #4
You're most welcome. And I learned too from digging in this (after all, not every physicist can offhandedly cough up what a 4th order CE eqn is, so I had to google too :wink: )
 

1. What is the Cauchy-Euler equation and what are repeated complex conjugate roots?

The Cauchy-Euler equation, also known as the homogeneous linear differential equation, is a second order differential equation with constant coefficients. It is in the form of ax^2y'' + bxy' + cy = 0, where a, b, and c are constants. Repeated complex conjugate roots refer to the case where the roots of the characteristic equation are complex numbers of the form a±bi, where a and b are real numbers, and they appear in pairs.

2. How do we solve for repeated complex conjugate roots in Cauchy-Euler equations?

To solve for repeated complex conjugate roots in Cauchy-Euler equations, we first find the roots of the characteristic equation, which will be of the form a±bi. Then, we use the corresponding solution formula y = e^(ax)(c1cos(bx) + c2sin(bx)) to find the general solution. However, in the case of repeated roots, we must also include the term xe^(ax) in the solution.

3. What is the significance of repeated complex conjugate roots in Cauchy-Euler equations?

The presence of repeated complex conjugate roots in Cauchy-Euler equations indicates that the general solution will contain complex-valued functions. This is important in applications where the solution represents a physical quantity, as it may indicate oscillatory or damped behavior.

4. Are there any special cases when dealing with repeated complex conjugate roots in Cauchy-Euler equations?

Yes, there are two special cases to consider when dealing with repeated complex conjugate roots. The first case is when the roots are purely imaginary (b=0), in which case the solution will be a linear combination of sine and cosine functions. The second case is when the roots are equal (a=0), in which case the solution will contain a logarithmic term.

5. How do repeated complex conjugate roots affect the stability of solutions in Cauchy-Euler equations?

Repeated complex conjugate roots can affect the stability of solutions in Cauchy-Euler equations, as they may introduce oscillatory or exponentially growing or decaying behavior. This can be analyzed further by examining the real parts of the roots and using the Routh-Hurwitz stability criterion.

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