Single Variable Calculus, Chapter 7, 7.2-1, Section 7.2-1, Problem 54

Determine what values of $\lambda$ for which $y = e^{\lambda x}$ satisfies the equation $y + y' = y''$.


$
\begin{equation}
\begin{aligned}

& \text{if } y = e^{\lambda x}, \text{ then }
\\
\\
& y' = e^{\lambda x} (\lambda) = \lambda e^{\lambda x}
\\
\\
& \text{then,}
\\
\\
& y'' = \lambda e ^{\lambda x} (\lambda) = \lambda^2 e^{\lambda x}
\\
\\
& \text{so..}
\\
\\
& y + y' = y''
\\
\\
& e^{\lambda x} + \lambda e^{\lambda x} = \lambda^2 e^{\lambda x}
\\
\\
& e^{\lambda x} (1 + \lambda) = \lambda^2 e^{\lambda x}
\\
\\
& 1 + \lambda = \lambda^2
\\
\\
& 0 = \lambda^2 - \lambda - 1
\\
\\
& \text{By using Quadratic Formula,}
\\
\\
& \lambda = \frac{1 + \sqrt{5}}{2} \text{ and } \lambda = \frac{1 - \sqrt{5}}{2}

\end{aligned}
\end{equation}
$

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