Trouble with first-order exact equation

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Discussion Overview

The discussion revolves around understanding first-order exact equations in the context of differential equations, specifically focusing on the application of the product rule and chain rule in differentiation. Participants explore the confusion surrounding the differentiation of a product involving a dependent variable.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • A participant expresses confusion about the differentiation of the expression 2xy [dy/dx] + y^2 = e^(2x) and its relation to the product (xy^2).
  • Another participant suggests using the chain rule for differentiation to clarify the misunderstanding.
  • A later reply emphasizes the importance of specifying the variable with respect to which differentiation is performed, noting that the derivative of y^2 with respect to x involves the term dy/dx.
  • The original poster acknowledges the misunderstanding and recognizes the need for more practice with the chain rule.

Areas of Agreement / Disagreement

Participants generally agree on the necessity of using the chain rule in this context, but the discussion reflects an individual learning process rather than a consensus on broader theoretical aspects.

Contextual Notes

The discussion highlights a limitation in the original poster's understanding of differentiation techniques, particularly in distinguishing between differentiation with respect to different variables and applying the chain rule correctly.

lukka
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Hi everyone, any help with this would be greatly appreciated..

I have been practicing simple differential equations for a couple months now and kinda just taking it easy and enjoying building my intuition. i have encountered a chapter in my text by Backhouse (pure mathematics 2) involving first order exact equations as a prelude to using integrating factors. It shows by example an inseparable differential equation..

2xy [dy/dx] +y^2 = e^(2x)

whose LHS is said to be equal to the derivative of the product (xy^2). The trouble I'm having here is that when i check and differentiate (xy^2) by product rule, i wind up with just 2xy + y^2. My question is where does the factor of [dy/dx] in the original equation come from? i suspect that i might be doing something wrong here?
 
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You must use chain rule.
 
lukka said:
Hi everyone, any help with this would be greatly appreciated..

I have been practicing simple differential equations for a couple months now and kinda just taking it easy and enjoying building my intuition. i have encountered a chapter in my text by Backhouse (pure mathematics 2) involving first order exact equations as a prelude to using integrating factors. It shows by example an inseparable differential equation..

2xy [dy/dx] +y^2 = e^(2x)

whose LHS is said to be equal to the derivative of the product (xy^2). The trouble I'm having here is that when i check and differentiate (xy^2) by product rule, i wind up with just 2xy + y^2. My question is where does the factor of [dy/dx] in the original equation come from? i suspect that i might be doing something wrong here?
Never say "differentiate" without specifying "differentiate with respect to which variable?"

Here, you are differentiating with respect to x. The derivative of y^2 with respect to x is NOT 2y. That is the derivative of y^2 with respect to y. The derivative of y^2 with respect to x is (by the chain rule that szynkasz mentions) 2y dy/dx.
 
I see where I'm going wrong, thanks to you both for pointing this out for me.. clearly need more practice with identifying when to use the chain rule! Thanks again!
 

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