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October 10th, 2015, 03:02 AM   #1
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2nd Order diff (PLZ HELP ! :D)

How do i approach this question??

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I got a) as my" + By' =mg
however can't solve for part b)
Any help would be greatly appreciated!!
Thank you!!!!!
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October 28th, 2015, 05:45 AM   #2
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This is a linear non-homogeneous differential equation with constant coefficients. Its associated homogeneous equation is $\displaystyle my''+ \beta y'= 0$ which has "characteristic equation" $\displaystyle mr^2+ \beta r= r(mr+ \beta)= 0$. The roots of that are $\displaystyle r= 0$ and $\displaystyle r= -\frac{\beta}{m}$. That means that the general solution to the associated homogeneous equation is $\displaystyle y(t)= C_1e^{0t}+ C_2e^{-\frac{\beta}{m}t}= C_1+ C_2e^{-\frac{\beta}{m}t}$.

Since the "non-homogeneous" part, -g, is constant, normally, we would try a constant, A, as a solution to the entire equation. But a constant is already a solution to the associated homogeneous equation so we try y= At instead. With y= At, y'= A and y''= 0 so the equation becomes $\displaystyle m(0)+ \beta(A)= -g$ and $\displaystyle A= -\frac{g}{\beta}$.

The general solution to the differential equation is $\displaystyle y(t)= C_1+ C_2e^{-\frac{\beta}{m}t}- \frac{g}{\beta}t$.
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