Seemingly Non-Separable Differential Equation

[danomite]

Homework Statement

I am trying to find the parametric equation that describes the following second order differential equation:

Homework Equations

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

Where m, g, and k are all constants.

The Attempt at a Solution

I substituted [itex]u=\frac{dy}{dt}[/itex] to reduce the order of the equation to one. Now I have:

[itex]m\frac{du}{dt}=-mg-ku[/itex]

And I have been stuck here. I don't see how to separate the variables, can anybody help out?

 

[micromass]

It's just a linear differential equation with constant coefficients. Have you seen how to solve these??

 

[danomite]

I have seen how to solve simple ones. My main problem here is isolating t and u. The algebra doesn't work out, and I've been so far unsuccessful in finding a substitution that will separate these variables.

 

[arildno]

Find a suitable CONSTANT addition to "u"

 

[HallsofIvy]

Homework Statement

I am trying to find the parametric equation that describes the following second order differential equation:

Homework Equations

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

Where m, g, and k are all constants.

The Attempt at a Solution

I substituted [itex]u=\frac{dy}{dt}[/itex] to reduce the order of the equation to one. Now I have:

[itex]m\frac{du}{dt}=-mg-ku[/itex]

And I have been stuck here. I don't see how to separate the variables, can anybody help out?

Since there is no variable, t, itself in the equation, it is pretty trivial to separate!

[itex]m\frac{du}{ku+ mg}= - dt[/itex]

Now integrate both sides.