Problem

$y^{\prime \prime}-y=0$

Answer

Expert–verified
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Answer

\(\boxed{y(x) = C_1 e^{-x} + C_2 e^{x}}\)

Steps

Step 1 ：Assume a solution of the form: \(y = e^{rx}\)

Step 2 ：Substitute into the equation and solve for r: \(r^2 - 1 = 0\)

Step 3 ：Find the values of r: \(r = -1, 1\)

Step 4 ：Write the general solution: \(y(x) = C_1 e^{-x} + C_2 e^{x}\)

Step 5 ：\(\boxed{y(x) = C_1 e^{-x} + C_2 e^{x}}\)

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