Single Variable Calculus Questions and Answers

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

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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} $