Integration Factors: Solving Problems

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Discussion Overview

The discussion revolves around finding integration factors for two differential equations. Participants explore methods for solving these equations, including assumptions about the form of the integrating factor and modifications to the equations themselves. The scope includes mathematical reasoning and problem-solving techniques related to differential equations.

Discussion Character

  • Mathematical reasoning
  • Homework-related
  • Exploratory

Main Points Raised

  • One participant presents two differential equations and expresses difficulty in finding the integration factor for both.
  • Another participant suggests assuming the integrating factor is a function of either x or y only and solving a differential equation to find it.
  • A participant describes their attempt to use a method from a book involving derivatives of M and N, but they are uncertain about their calculations and whether they correctly derived an integrating factor.
  • The second equation is mentioned as potentially solvable by inspection, though the reasoning behind this is questioned by another participant.
  • Participants reference expected solutions for both equations without confirming their correctness.
  • One participant later indicates they have resolved their issue without elaborating on the solution.

Areas of Agreement / Disagreement

The discussion contains multiple viewpoints on how to approach finding the integration factors, with no consensus reached on the methods or solutions for the equations presented. Participants express uncertainty and differing opinions on the solvability of the second equation by inspection.

Contextual Notes

Participants do not clarify the assumptions underlying their methods or the specific modifications needed for the second equation. There is also a lack of resolution regarding the correctness of the proposed solutions.

confusedM
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I am having issues finding the integration factor for the following two problems. I believe the second one can be solved by inspection.

1. (y^3+2ye^x)dx + (e^x+3y^2)dy = 0


2. (x-x^2-y^2)dx + (y+x^2+y^2)dy = 0
 
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For the first one, try assuming in turn that the integrating factor is a function of either x or y only, then see if you can come up with it by solving a DE.

What do you mean by 2nd one being solvable by inspection?
 
I tried using the method described by the book. Setting M= y^3+2ye^x and N = (e^x+3y^2)... then taking dM/dy and dN/dx... then (dM/dy - dN/dx) / N and (dN/dx - dM/dy) / M... I don't know if my problem is that I did something wrong in the derivations but neither gives me an integration factor that is a function of just y or just x.

As far as the second goes, I meant that the equation needs to be modified so that the above is possible (dM/dy = dN/dx), but I can't figure out how to modify the equation around so that is possible.

The solutions are supposed to be:
1. y^3e^x+ye^(2x)=c
2. ln(x^2 + y^2) +2y - 2x = c
 
never mind. I got it.
 

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