# Trouble with first-order exact equation

## Main Question or Discussion Point

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.

HallsofIvy
Homework Helper
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!