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Given the following system of two differential equations.

$$\frac{dx}{dt}=\alpha(y-x)$$

$$\frac{dy}{dt}=\alpha(x-y)$$

Reduce the above system of two differential equation to a system of one dimension differential equation and solve the resulting one-dimensional differential equation.

What i tried

I tried relating both the system of differential equation, $\frac{dx}{dt}=-\frac{dy}{dt}$ Then integrating the following expression i got, $x(t)=-y(t)+c$. Am i correct. Could anyone please explain. Thanks

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  • $\begingroup$ How do u get the one-dimensional differential equation first? $\endgroup$ – ys wong Apr 23 '17 at 18:45
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You can write the system this way $$ \frac{d}{dt} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -\alpha & \alpha \\ \alpha& -\alpha \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} $$ It then is possible to interpret this system as one vector equation, which has one vector variable, however that vector is two dimensional, which does not fit your wording. $$ \dot{u} = A u $$

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  • $\begingroup$ So i must reduced the system of equations to a matrix form ? And the matrix form is considered a one-dimension system. Then next i must solve the matrix, by getting the eigenvalues/eigenvectors etc? $\endgroup$ – ys wong Apr 23 '17 at 16:30
  • $\begingroup$ You should check your wording, I believe there is some mix up between variables and dimensions. $\endgroup$ – mvw Apr 23 '17 at 16:34

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