Can homogeneous substitution solve this differential equation?

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Homework Help Overview

The discussion revolves around solving a differential equation of the form x(dy/dx) - y = sqrt(xy + x^2). The original poster attempts to understand the manipulation of terms leading to a specific form of the derivative.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the transformation of the equation using the substitution u = y/x and question the algebraic steps involved in simplifying the square root and fraction.

Discussion Status

Some participants have offered explanations regarding the algebraic manipulation, while others express confusion about the steps taken. The conversation indicates a collaborative effort to clarify the reasoning behind the transformations without reaching a definitive conclusion.

Contextual Notes

There is an acknowledgment of potential gaps in foundational algebra knowledge, which may affect understanding of the current problem. The original poster also notes formatting issues in their initial post.

jdawg
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Homework Statement


x(dy/dx) - y = sqrt(xy +x2)

Homework Equations

The Attempt at a Solution


I got up to this point: u=y/x

dy/dx = (sqrt(xy+x2))/x + y/x

And then the solution shows this:

dy/dx = y/x + (y/x+1)½

Please help, I don't understand how they got to that point.
 
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Also, sorry about the bad formatting. Where did they move the square root symbol and fraction symbol?
 
It's fairly elementary algebra that was probably dinned into you at some point and you have forgotten.

√(xy + x2)/x = √(xy + x2)/√((x2) = √[(xy + x2)/x2] = ...

You might last time have lost sight of the why, just drilled to do that: hope something comes back but this time instead look to see how it makes sense.
 
Last edited:
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Thank you so much! Haha its been a minute since I've had algebra. Great explanation! :D
 

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