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Hw$08-2\mathrm{.}3-$SODE$-$CCE: Problem $18$

Problem Value: $10$ point$($s$).$ Problem Score $75\%.$ Attempts Remaining: $16$ attempts.

Help Entering Answers See Example $2\mathrm{.}3\mathrm{.}10,$ in Section $2\mathrm{.}3,$ in the MTH $235$ Textbook

$(10$ points$)$

Homework: Homogeneous Constant Coefficients

An object of mass $2$ grams is attached to a vertical spring with spring constant $32$ grams $\frac{?}{se{c}^2}.$ Neglect any friction with the air.

$($a$)$ Find the differential equation $y^{\text{'}\text{'}}=f(y,y^{\text{'}})$ satisfied by the function $y,$ the displacement of the object from its equilibrium position, positive downwards. Write $y$ for $y(t)$ and yp for $y^{\text{'}}(t).$

$f(y,y^{\text{'}})=$

$($b$)$ Find $r_1,r_2,$ roots of the characteristic polynomial of the equation above.

$r_1,r_2=$

$($b$)$ Find a set of real$-$valued fundamental solutions to the differential equation above.

$y_1(t)=,$

$y_2(t)=,$

$($c$)$ At $t=0$ the object is pulled up $2\sqrt[\phantom{2}]{3}cm$ and the released with an initial velocity downwards of $8c\frac{m}{sec}.$ Find the amplitude $A>0$ and the phase shift $in(-,]$ of the subsequent movement.

$A=,=,$