Problem

\( y^{\prime \prime}-y^{\prime}-y=0 \)

Answer

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Answer

\( y(x) = C_1e^{\frac{1 + \sqrt{5}}{2}x} + C_2e^{\frac{1 - \sqrt{5}}{2}x} \)

Steps

Step 1 ：\( m^2 - m - 1 = 0 \)

Step 2 ：\( m = \frac{1 \pm \sqrt{5}}{2} \)

Step 3 ：\( y(x) = C_1e^{\frac{1 + \sqrt{5}}{2}x} + C_2e^{\frac{1 - \sqrt{5}}{2}x} \)

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