Substitution Differential Equation

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Homework Help Overview

The problem involves solving an initial value problem represented by a differential equation of the form ydx + x(lnx - lny - 1)dy = 0. The subject area pertains to differential equations, specifically methods of substitution and homogeneity.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster expresses confusion regarding the applicability of substitution methods due to the non-homogeneous nature of the equation. Some participants suggest exploring the substitution u = x/y, while others question the homogeneity of the equation, noting the relationship between ln(x) and ln(y).

Discussion Status

The discussion is ongoing, with participants exploring different substitution approaches and questioning the assumptions about homogeneity. Some guidance has been offered regarding potential substitutions, but no consensus has been reached.

Contextual Notes

Participants are navigating the constraints of the problem, particularly regarding the definitions of homogeneity and the suitability of various substitution methods.

KillerZ
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Homework Statement



Solve the given initial value problem.

Homework Equations



[tex]ydx + x(lnx - lny - 1)dy = 0[/tex]

The Attempt at a Solution



I am confused as to what to do as I can't just substitute y=ux or x=vy because they are not homogeneous of the same degree. And I can't use exact because:

[tex]\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}[/tex]
 
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I don't think u=x/y is such a bad start. ln(x)-ln(y)=ln(x/y)=ln(u). I haven't tried to follow it out. But I would definitely start with that.
 
Are you sure it isn't homogeneous? Remember ln(x) - ln(y) = ln(x/y)
 
I got it thanks for the help.
 

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