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[itex]y=c_1e^t+c_2e^{-t}[/itex]

With t = 0, y = 2 we have [itex]c_1 + c_2 = 2[/itex]

Upon differentiation of y, we obtain: [itex]y'=c_1e^t-c_2e^{-t}[/itex]

With t = 0, y' = -1 [itex]c_1 - c_2 = -1[/itex]

Now the book says that upon solving these two equations simultaneously, [itex] c_1 = 1/2 & c_2 = 3/2[/itex]

So I try to verify the result, I set up the two equations to solve by elimination

[itex]c_1 + c_2 = 2[/itex]

[itex]c_1 - c_2 = -1[/itex]

c_2 cancels out & we obtain c_1 = 1 which is the wrong conclusion. I can reason out that c_1 should be 1/2 & c_2 should be 3/2, but first and foremost I want to know why solving by elimination gives the wrong answer and what method would be the correct one.