Anyone know how to solve these equations without using eigenvalues/eigenvectors? Can these be solved using standard integration techniques? dx/dt = 4x+2y dy/dt = -x +y where x and y are functions of t.
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$\displaystyle \frac{d^{\, 2}x}{dt^{2}}=4\frac{dx}{dt}+2\frac{dy}{dt} = 4\frac{dx}{dt}-2x+2y = 4\frac{dx}{dt}-2x+\left(\frac{dx}{dt}-4x\right)$ so $\displaystyle \frac{d^{\, 2}x}{dt^{2}} - 5\frac{dx}{dt}+6x = 0,$ etc.
Many thanks... Seems obvious now!
You will need to do something similar to get a DE for y as well.
Still working on this... I am now trying to understand why the eigenvalues of the matrix of coeff on the RHS are the powers for the solution to the auxilliary equation? Can anyone help?
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