# I have a system of coupled differential equations (an example )in the form of, x &#x203

juanberrio8a 2022-06-24 Answered
I have a system of coupled differential equations (an example )in the form of,
${x}^{″}+a{x}^{\prime }+bx-cy=0$
${y}^{″}-a{y}^{\prime }+by-cx=0$
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 $\frac{f\left(w\right)}{g\left(w\right)}$ 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?
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## Answers (1)

Anika Stevenson
Answered 2022-06-25 Author has 19 answers
Regarding the linear system of ode's, applying the Laplace transform on
$\left\{\begin{array}{l}{x}^{″}+a{x}^{\prime }+bx-cy=0\\ {y}^{″}-a{y}^{\prime }+by-cx=0\end{array}$
we have
$\left\{\begin{array}{l}\left({s}^{2}+as+b\right)X\left(s\right)=cY\left(s\right)+{c}_{1}{x}_{0}+{c}_{2}{\stackrel{˙}{x}}_{0}\\ \left({s}^{2}-as+b\right)Y\left(s\right)=cX\left(s\right)+{c}_{3}{y}_{0}+{c}_{4}{\stackrel{˙}{y}}_{0}\end{array}$
and solving for $X\left(s\right)$ we have
$X\left(s\right)=-\frac{\left(s\left(s-a\right)+b\right)\left({c}_{1}{x}_{0}+{c}_{2}{\stackrel{˙}{x}}_{0}\right)+c\left({c}_{3}{y}_{0}+{c}_{4}{\stackrel{˙}{y}}_{0}\right)}{{a}^{2}{s}^{2}-{\left(b+{s}^{2}\right)}^{2}+{c}^{2}}$
now focusing on the denominator, the condition to have only real roots is
${a}^{2}-2b±\sqrt{{a}^{4}-4{a}^{2}b+4{c}^{2}}\ge 0$
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