Trouble with first-order exact equation

1. Jul 5, 2013

lukka

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?

2. Jul 5, 2013

szynkasz

You must use chain rule.

3. Jul 6, 2013

HallsofIvy

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.

4. Jul 6, 2013

lukka

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!