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I'm having a difficult time finding the coefficients of these set of equations:

$A_1 + A_2 + A_3 + A_4 = 0 $
$A_1 + A_2 - A_3 - A_4 = 0 $
$A_1e^{\pi/2} + A_2e^{-\pi/2} - A_3e^{j\pi/2} - A_4e^{-j\pi/2}=1 $
$-8A_3A_4 + 8A_1A_2 = 1$

Someone suggested I convert the $e^j$ terms to sine/cosines, but I got a weird simplification that was 1=0....Can someone give me any tips? The answers should result in the following equation:

$x(t) = A_1e^{t} + A_2e^{-t} + A_3e^{jt} + A_4e^{-jt}$

Containing all real numbers

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Here's one tip:

Since all the $A_i$ are real numbers, and $e^{j\pi/2}$ and $e^{-j\pi/2}$ are conjugates and the only non-real numbers in the third equation, $A_3$ and $A_4$ must be opposites (negatives) of each other. Otherwise, the left side of the equation would have a non-zero imaginary part that the right side does not have.

That reduces the number of variables by one.

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