Solving DE: Integrating Factor Method

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Homework Help Overview

The discussion revolves around solving a differential equation using the integrating factor method. The original poster presents the equation dy/dx = (-2x + 5y)/(2x + y) and seeks guidance on finding an integrating factor and solving it as an exact equation.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Some participants explore the idea of defining a new variable, z = y/x, to simplify the equation. Others discuss the transformation of the equation and the resulting expressions, questioning the signs in their calculations.

Discussion Status

The discussion is active, with participants sharing their approaches and results. There is a recognition of differing outcomes in calculations, prompting suggestions for rechecking work. No consensus has been reached yet.

Contextual Notes

Participants are navigating the complexities of the differential equation and the implications of their variable transformations. There is an emphasis on ensuring accuracy in their mathematical manipulations.

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What approach should be used to solve the following DE:

dy/dx= (-2x+5y)/(2x+y)

Find an integrating factor and solve it as an excact equation?

Thanks.
 
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See what happens if you define a new variable: z = y/x
 
Thanks!

Right, I get

1/[(-2+5z)/(2+z)-z] dz = (1/x) dx

so

3ln(z-1) -4ln(z-2) = lnx +c

and

3ln(y/x -1) -4ln(y/x-2)-lnx = c

?
 
Good - but I got different signs on my first try. You should recheck.
 

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