Solving a system of First Order ODE's

AlexKolimbarides

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iv) Which of Y1, Y2, Y3, and Y4 is the most general solution Y t = (y1, y2) to the following linear system?

. . . . .\(\displaystyle \dfrac{dy^1}{dx}\, +\, 3\, \dfrac{dy^2}{dx}\, =\, -4y^1\, -\, 6y^2,\). . .\(\displaystyle 2\, \dfrac{dy^1}{dx}\, +\, \dfrac{dy^2}{dx}\, =\, 7y^1\, +\, 8y^2\)



. . . . .\(\displaystyle Y_1:\, c_1\, e^{2x}\, \dbinom{0}{1}\, +\, c_2\, e^{-2x}\, \dbinom{-1}{1},\). . . . .\(\displaystyle Y_2:\, c_1\, e^{2x}\, \dbinom{2}{-1}\, +\, c_2\, e^{-1}\, \dbinom{-1}{1}\)

. . . . .\(\displaystyle Y_3:\, c_1\, e^{-3}\, \dbinom{2}{-1}\, +\, c_2\, e^{-2x}\, \dbinom{1}{1},\). . . . .\(\displaystyle Y_4:\, c_1\, e^{-2x}\, \dbinom{-2}{-1}\, +\, c_2\, e^{-3x}\, \dbinom{1}{-2}\)



. . . . .(A) Y1 . . . . .(B) Y2 . . . . .(C) Y3 . . . . .(D) Y4 . . . . .(E) none of these



I'm not sure how well you can see the question above, but its basically a system of first order ODE's with 2 variables y1 and y2

dy1/dx + 3dy2/dx = -4y1 - 6y2 and 2dy1/dx + dy2/dx = 7y1 + 8y2

And I need to find the most general solution Y^t = (y1, y2)

I've attempted to get the eigenvalues of this but have somehow come to the answer that they are imaginary, which i don't think is correct.
I also have no answer for this question as my lecturer has not uploaded any answers to past papers.

Thank you in advance
 

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