Seemingly Non-Separable Differential Equation

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Homework Help Overview

The discussion revolves around a second order differential equation involving constants m, g, and k, specifically focusing on finding a parametric equation. The equation is presented in the form of a linear differential equation with constant coefficients.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to reduce the order of the differential equation by substituting u for dy/dt, leading to a first-order equation. They express difficulty in isolating the variables t and u, questioning the algebraic manipulation required for separation.
  • Some participants suggest that the equation is linear and may not require separation in the traditional sense, prompting further exploration of potential substitutions or constants that could facilitate the solution.

Discussion Status

Participants are actively engaging with the problem, with some offering guidance on possible approaches, such as integrating after a suitable substitution. There is an acknowledgment of differing interpretations regarding the separation of variables, indicating a productive exploration of the problem.

Contextual Notes

There is a noted absence of the variable t in the transformed equation, which some participants suggest may simplify the separation process. The original poster's struggle with algebraic manipulation highlights potential assumptions about the nature of the equation.

danomite
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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?
 
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It's just a linear differential equation with constant coefficients. Have you seen how to solve these??
 
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.
 
Find a suitable CONSTANT addition to "u"
 
danomite said:

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.
 

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