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(a) For what values of $ r $ does the function $ y = e^{rx} $ satisfy the differential equation $ 2y^{"} + y^{'} - y = 0? $ (b) if $ r_1 $ and $ r_2 $ are the values of $ r $ that you found in part (a), show that every member of the family of functions $ y = ae^{r_1{x}} + be^{r_2{x}} $ is also a solution.

The drawing (not to scale) shows one alignment of the sun, earth, and moon. The gravitational force $\overline{\mathbf{F}}_{\text { SM }}$that the sun exerts on the moon is perpendicular to the force $\overrightarrow{\mathbf{F}}_{\mathrm{EM}}$ that the earth exerts on the moon. The masses are: mass of sun $=1.99 \times 10^{30} \mathrm{kg}$ , mass of earth $=5.98 \times 10^{24} \mathrm{kg}$ , mass of moon $=7.35 \times 10^{22} \mathrm{kg}$ . The distances shown in the drawing are $r_{\mathrm{SM}}=1.50 \times 10^{11} \mathrm{m}$ and $r_{\mathrm{EM}}=3.85 \times 10^{8} \mathrm{m} .$ Determine the magnitude of the net gravitational force on the moon.

Evaluate $ f(-3) $ , $ f(0) $ and $ f(2) $ for the piecewise defined function. Then sketch the graph of the function. $ f(x) = \left\{ \begin{array}{ll} 3 - \frac{1}{2}x & \mbox{if $ x < 2 $}\\ 2x - 5 & \mbox{if $ x \ge 2 $} \end{array} \right.$

Two fun-loving otters are sliding toward each other on a muddy (and hence frictionless) horizontal surface. One of them, of mass 7.50 $\mathrm{kg}$ , is shiding to the left at 5.00 $\mathrm{m} / \mathrm{s}$ , while the other, of mass 5.75 $\mathrm{kg}$ , is slipping to the right at 6.00 $\mathrm{m} / \mathrm{s}$ . They hold fast to each other after they collide. (a) Find the magnitude and direction of the velocity of these free-spirited otters right after they collide. (b) How much mechanical energy dissipates during this play?