My Math Forum 2nd Order diff (PLZ HELP ! :D)

 Differential Equations Ordinary and Partial Differential Equations Math Forum

 October 10th, 2015, 03:02 AM #1 Newbie   Joined: Sep 2015 From: AU Posts: 5 Thanks: 0 2nd Order diff (PLZ HELP ! :D) How do i approach this question?? Imgur: The most awesome images on the Internet Imgur: The most awesome images on the Internet (it's not letting me embed the image so here are the links) I got a) as my" + By' =mg however can't solve for part b) Any help would be greatly appreciated!! Thank you！！！！！
 October 28th, 2015, 05:45 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 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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