Why Are There Different Methods for Solving Differential Equations?

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Discussion Overview

The discussion centers around the methods for solving differential equations, specifically exploring why different approaches yield different results. Participants examine a specific problem involving the relationship between position, velocity, and time, and the implications of treating variables as constants during integration.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the order of solving a differential equation and requests clarification on the integration process.
  • Another participant asserts that neither velocity (v) nor time (t) can be treated as constants, emphasizing that v depends on t.
  • A participant suggests that integration should be performed without treating variables as constants, highlighting the importance of recognizing when variables are functions of one another.
  • It is noted that separating variables is only valid if the equation is separable; otherwise, different methods must be employed.
  • A later reply acknowledges the helpfulness of the explanations provided in the discussion.

Areas of Agreement / Disagreement

Participants generally agree on the importance of not treating variables as constants when they are not, but there is no consensus on the best approach to solving the specific differential equation discussed. The discussion remains unresolved regarding the implications of different methods used.

Contextual Notes

Participants express uncertainty about the integration steps and the treatment of variables, indicating potential limitations in their understanding of the problem and the methods applicable to it.

LENIN
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I found this exercise in an old physics schoolbook. I maeged to solve it but I don't really understand why I have to solve it in this excact order. Before I start I would just like to add that I have very little experience with differential equations (they are not in our high school year plan), and therefor any explanations will be welcomed.

So here it goes.

t=?
s=10
v=2s+1
v=ds/dt
dt=(1/v)ds
t=Iteg[1/v]ds
t=1/2Ln(2s+1)
This is the way it was solved in the book (I might have memorized the integration of v wrong).

But I tried to solve it like this
t=?
s=10
v=2s+1
v=ds/dt
ds=v*dt
s=(2s+1)t
s/(2s+1)=t

The solutions are clearly different. I don't understend why. The only idea I got was that I can't integrate v by dt becouse t is not a veriable in v and v is therefore threted as a constent what changes the whole thing (is this true). But if there are any beter explanations I would be glad to hear them. Thanks.
 
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Neither v nor t is a constant here nor can be treated as one. When you write
"ds= v dt so s= (2s+1)t" you are treating v as a constant- and it is not. It depends upon t. Since you don't yet know v or s as a function of t, the only way to integrate
ds= (2s+1)dt is as ds/(2s+1)= dt which is exactly the book's solution.
 
So in general I should always integrate in such a way that the expresion is not threated as a constant, if posibel of course. Right.
 
Not just "if possible". You should not assume a variable is a constant when it is not!

What you are doing there is separating the variables- so that you have only s on one side of the equation and t on the other. If that is NOT possible then it is not a "separable" equation and you will have to use some other method. If x is a function of t then you cannot integrate [tex]\int f(x,t)dt[/tex] without knowing exact what function it is!
 
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Thank you! Your explanation is really very helpful. :smile:
 

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