Test for Exactness, linearly dependent?

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Discussion Overview

The discussion revolves around the topic of testing for exactness in differential equations, specifically focusing on the equation (2x + y^2)dx + 4xydy = 0. Participants explore the implications of linear dependence and independence of solutions, the search for integrating factors, and the process of solving the equation.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions the relevance of linear dependence in testing for exactness, asserting that the equation is not exact because the partial derivatives are not equal.
  • Another participant proposes that since the difference between the partial derivatives is a power of y, an integrating factor that is a power of x might be worth exploring.
  • A participant reports finding an integrating factor of 1/sqrt(x) but notes that it still leads to a non-exact solution, expressing confusion over the relationship between M/y and N/x.
  • Another participant confirms that 1/sqrt(x) is indeed a valid integrating factor and provides a reformulation of the equation after applying it, suggesting a function F(x,y) that could be derived from the modified equation.
  • A later reply acknowledges an earlier error in differentiation and confirms that the integrating factor worked, leading to a successful solution.

Areas of Agreement / Disagreement

Participants express differing views on the relevance of linear dependence in the context of exactness. While some assert that the equation is not exact, others explore the implications of integrating factors and their application. The discussion remains somewhat unresolved regarding the initial confusion over the relationship between the terms after applying the integrating factor.

Contextual Notes

There are indications of missing assumptions or errors in differentiation that may have influenced the participants' understanding of the problem. The discussion also reflects varying interpretations of the role of linear dependence in the context of exactness.

ombudsmansect
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Hey guys was wondering if anyone knew what the go is with linearly dependent solutions to test for exactness, by that I mean
I have the differential equation (2x + y^2)dx + 4xydy = 0 (M,N)
So i test for exactness and

\partialM/\partialy = 2y \partialN/\partialx = 4y

So I wanted to ask if this does prove exactness, given that they are linearly dependent I cannot find an integrating factor, but I am not sure if I can just proceed to solve anyway
 
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"linearly independent" or "dependent" has nothing to do with this problem. The given equation is NOT exact because 4y and 2y are not equal. Nor does "dependent" or "independent" have anything to do with you not being able to find an integrating factor.

However, the fact that N_x- M_y is a power of y only might suggest that you look for an integrating factor that is a power of x only: x^n.
 
hello again halls thanks for looking again,

I have run through the process to find an integrating factor and it gives me 1/sqrtx. This is also an intuitive guess at an integrating factor but it still yields a non exact solution, even though the formula specifically yields this answer. With integrating fact 1/sqrtx the test yields:

M/y = (2y)/sqrtx N/x=2y

there seems to be something about this relationship that says i cannot equate them. Perhaps there is another way I cannot see
 
1/sqrt(x) is an integrating factor test again
 
ombudsmansect said:
hello again halls thanks for looking again,

I have run through the process to find an integrating factor and it gives me 1/sqrtx. This is also an intuitive guess at an integrating factor but it still yields a non exact solution, even though the formula specifically yields this answer. With integrating fact 1/sqrtx the test yields:

M/y = (2y)/sqrtx N/x=2y

there seems to be something about this relationship that says i cannot equate them. Perhaps there is another way I cannot see
Then you have an error in your solution. x^{-1/2}= 1/\sqrt{x} certainly is an intgrating factor. Multiplying your equation by that,
(2x^{1/2} + x^{-1/2}y^2)dx + 4x^{1/2}ydy = 0

M_y= 2x^{-1/2}y
and
N_x= 4(1/2)x^{-1/2}y= 2x^{-1/2}y

Now, that means that there exist a function, F(x,y), such that
F_x= 2x^{1/2}+ x^{-1/2}y^3
so that
F= \frac{4}{3}x^{3/2}+ 2x^{1/2}y^3+ \phi(y)

Can you finish it?
 
hey guys thanks for your help, it most certainly was an integrating factor as you say, some careless differentiation there on my part. Worked out very nicely in the end with k(y) as 0 so the answer is just 4/3 x^1.5 + 2x^1.5y^3 as you have stated there hallsofivy. thanks again guys
 

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