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amolv06

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## Homework Statement

Find an equation describing a body in free-fall in air. This is not really homework. I'm in a differential equations class write now, and I have fun finding real world applications for such things.

## Homework Equations

[tex]m\frac{d^{2}y}{dt^{2}} = B\frac{dy}{dt}-mg[/tex]

## The Attempt at a Solution

Rewriting the above:

[tex]\frac{d^{2}y}{dt^{2}} - \frac{b}{m}\frac{dy}{dt} = -g[/tex]

The corresponding homogeneous equation is:

[tex]\frac{d^{2}y}{dt^{2}} - \frac{b}{m}\frac{dy}{dt} = 0[/tex]

Two solutions to the homogeneous equations are:

[tex]y_{1} = C_{1}[/tex] and [tex]y_{2} = C_{2}e^{\frac{b}{m}t}[/tex]

And as a particular solution:

[tex]y_{p} = \frac{gm}{b}t[/tex]

Therefore by the superposition principle, we have a general equation as follows:

[tex]y(t) = y_{1}(t) + y_{2}(t) + y_{p}(t) = C_{1} + C_{2}e^{\frac{b}{m}t} + \frac{gm}{b}t[/tex]

So here's where my question arises. I believe my assumption (the initial differential equation) should be a reasonable approximation of the real world. And I plugged my solution into the original differential equation. I seem to have answered it correctly. If both of these assumptions are true, then this equation doesn't make sense. It asserts that we are falling up! The only we I can think that this equation can still hold is if B were negative. I think after that, everything should make sense. Is this a correct assumption, or did I just make a mistake in the mathematics above?

Thanks in advance for your time and any help.

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