I have a system of coupled differential equations (an example )in the form of,

${x}^{\u2033}+a{x}^{\prime}+bx-cy=0$

${y}^{\u2033}-a{y}^{\prime}+by-cx=0$

The solution to the above system looks like,

$x=A{e}^{{w}_{1}t}+B{e}^{{w}_{2}t}+C{e}^{{w}_{3}t}+D{e}^{{w}_{4}t}$

$y=E{e}^{{w}_{1}t}+F{e}^{{w}_{2}t}+G{e}^{{w}_{3}t}+H{e}^{{w}_{4}t}$

the frequencies ${w}_{1}$,${w}_{2}$,${w}_{3}$ and ${w}_{4}$ are functions of a,b and c My goal is to determine that values of a,b and c such that all w are Real. However, my original system of differential equations has nonlinear terms. I managed to derive the solution but in power expansion form or,

$x=\sum _{i=0}^{n}\frac{{a}_{i}}{i!}{t}^{i}$

$y=\sum _{i=0}^{n}\frac{{a}_{i}}{i!}{t}^{i}$

I need to determine the frequencies so that I can find the values of a,b and c such that all w are Real. Is it possible to derive w from limited power expansion (say 8th term) of x and y?

My first attempt:

I did a test of my method by taking Fourier Sine Transform of analytical solution which give me the answer in form of $\frac{f(w)}{g(w)}$ where w is the frequency. Then solved the g(w) for w which gives me four answers for w which are the frequencies ${w}_{1}$,${w}_{2}$,${w}_{3}$ and ${w}_{4}$. However when I attempted with Fourier Sine Transform of series solutions, the answer is different. This is due to the limited power expansion. Is here a way to improve this?

My second attempt:

I linearize the nonlinear terms of differential equations and used the matrix to calculate Determinant of (A-wI) where A is the matrix of the system of differential equation. I managed to calculate the values of a,b and c but they were incorrect because of linear terms I made.

A last method that I am considering is Monte Carlo. Is it possible to get a frequency equation from limited power expansion of differential equation solution? Any ideas for other methods that I missed?

${x}^{\u2033}+a{x}^{\prime}+bx-cy=0$

${y}^{\u2033}-a{y}^{\prime}+by-cx=0$

The solution to the above system looks like,

$x=A{e}^{{w}_{1}t}+B{e}^{{w}_{2}t}+C{e}^{{w}_{3}t}+D{e}^{{w}_{4}t}$

$y=E{e}^{{w}_{1}t}+F{e}^{{w}_{2}t}+G{e}^{{w}_{3}t}+H{e}^{{w}_{4}t}$

the frequencies ${w}_{1}$,${w}_{2}$,${w}_{3}$ and ${w}_{4}$ are functions of a,b and c My goal is to determine that values of a,b and c such that all w are Real. However, my original system of differential equations has nonlinear terms. I managed to derive the solution but in power expansion form or,

$x=\sum _{i=0}^{n}\frac{{a}_{i}}{i!}{t}^{i}$

$y=\sum _{i=0}^{n}\frac{{a}_{i}}{i!}{t}^{i}$

I need to determine the frequencies so that I can find the values of a,b and c such that all w are Real. Is it possible to derive w from limited power expansion (say 8th term) of x and y?

My first attempt:

I did a test of my method by taking Fourier Sine Transform of analytical solution which give me the answer in form of $\frac{f(w)}{g(w)}$ where w is the frequency. Then solved the g(w) for w which gives me four answers for w which are the frequencies ${w}_{1}$,${w}_{2}$,${w}_{3}$ and ${w}_{4}$. However when I attempted with Fourier Sine Transform of series solutions, the answer is different. This is due to the limited power expansion. Is here a way to improve this?

My second attempt:

I linearize the nonlinear terms of differential equations and used the matrix to calculate Determinant of (A-wI) where A is the matrix of the system of differential equation. I managed to calculate the values of a,b and c but they were incorrect because of linear terms I made.

A last method that I am considering is Monte Carlo. Is it possible to get a frequency equation from limited power expansion of differential equation solution? Any ideas for other methods that I missed?