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ODE Integrating factor

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K

Thread Starter

kokkie_d

Joined Jan 12, 2009
72
Hi,

Could someone explain to me why the following step is allowed?
\(
e^{x}\frac{dy}{dx} + e^{x}y = e^{x}x
\)
\(
\frac{d}{dx}[ e^{x}y] = e^{x}x
\)

cheers,
 
Georacer

Georacer

Joined Nov 25, 2009
5,182
I think the chain rule has nothing to do here. It's just the expansion of the term
\((e^x \cdot y)'=(e^x)'\cdot y+e^x \cdot y'=e^x \cdot y+e^x \cdot y'\)
 
K

Thread Starter

kokkie_d

Joined Jan 12, 2009
72
@georacer:

Risking sounding silly here: What do you mean "expaning of the term"?
I mean I am familiar with finding zeros by expanding term as in:
\(
x^{2}-9 = (x+3)\cdot(x-3)
\)
but I am not sure how this applies here.

Please elaborate.
 
bradstormer

bradstormer

Joined Aug 6, 2010
65
i think the reason it is allowed is because its the same as multiplying fractions, for example

2/3*6 = (2*6)/3 = 12/3 = 4

think of the "d" part of the equation as just a function, just as if the equation was

-2/3*6 = -(2*6)/3 = -12/3 = -4

hope this was of some use :)
 
Georacer

Georacer

Joined Nov 25, 2009
5,182
Nonononono! Nothing of the sort!

There is an identity in differential analysis that says that given two functions f and g which are differentiable, the followin is valid:
\(\frac{d}{dx}(f(x) \cdot g(x))=\frac{df(x)}{dx} \cdot g(x) + f(x) \cdot \frac{dg(x)}{dx}\)

Is that clear?
 
K

Thread Starter

kokkie_d

Joined Jan 12, 2009
72
Yes, thank you.

I have found where it came from (had to revert to manual for dummies) lol

If you take the derivative of: \(e^{x}y\)
i.e. \(\frac{d}{dx}[e^{x}y]\)

you get:
\(e^{x}\frac{dy}{dx}+e^{x}y\)

I was looking way to far out.
 
S

someonesdad

Joined Jul 7, 2009
1,583
Georacer said:
I think the chain rule has nothing to do here.
Click to expand...
Georacer, I was going to respond angrily -- then I realized that my feeble brain mixed up the product rule with the chain rule. Thanks for correcting me. My only excuse is that I took calculus before most of you were born, so I can claim feeble-mindedness and forgetfulness. :p Of course, my wife would just claim it's overall incompetence. :D
 
Georacer

Georacer

Joined Nov 25, 2009
5,182
Well, if your wife competes with you in math, then you have much more serious issues that mathematical mixups...

No harm done, be sure of that. The time will come when you will correct me in electronics, I 'm sure of that.

Have a nice evening!
 
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