# Section 3.1 – Homogeneous Equations with Constant Coefficients

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
1. Solve the initial value problem $y''-y'-2y=0,\qquad y(0)=\alpha,\quad y'(0)=2.$
Find $$\alpha$$ so that the solution approaches zero as $$t\rightarrow \infty.$$

$$y=\dfrac{1}{3}(\alpha +2)e^{2t}+\dfrac{2}{3}(\alpha-1)e^{-t}$$
If $$\alpha=-2$$, then $$y=-2e^{-t}\to 0$$ as $$t\to \infty$$

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