Proof of Integration Factor for diff eq

[Ald]

Homework Statement

Looking for help on this Exact first order differential equation problem. Thanks

Show that if ((partial M/partial t)-(partial N/partial y))/M=Q(y)

then the differential equation M(t,y)+N(t,y)dy/dt=0 has an integrating factor

μ(y)=exp(integral Q(y)dy).

Homework Equations

The Attempt at a Solution

 

[Dick]

If mu(y) is an integrating factor, then mu(y)*M(t,y)*dt+mu(y)*N(t,y)*dy=0 is exact. Is it? Try the usual test for exactness. BTW, be clear where the y dependence is in mu(y). It's not in the integration variable, that's just a dummy. y is one of the limits of integration.

 

[Ald]

Thanks for your help, yes your recommendation makes sense.

I assume these are the partial derivatives I take, there is no actual expression to take the partial of, how do I prove exactness?

partial(mu(y)*M(t,y))/partial dy=partial (mu(y)*N(t,y))/partial dt

 

[Dick]

Use your formula for mu(y). The partial derivative of mu(y) wrt to y can be written in a different way. The partial derivatives of M and N, you just write as partial derivatives. You are trying to show that mu is an integrating factor leads to the expression in the 'if' part of your statement.

 

[Ald]

I'm sorry what does 'wrt' mean?

 

[Dick]

wrt='with respect to'. What's the derivative of mu(y) with respect to y?

 

[Ald]

The only other way I could find to write the mu(y) was in the form mu(t,y)=1/(xM-yN) is this what you had in mind?
Thanks

 

[Ald]

Thanks Dick for getting me in right direction, I think I have it proved.
Thanks again

 

[Dick]

Thanks Dick for getting me in right direction, I think I have it proved.
Thanks again

Did it really work? You differentiated the exponential and used the chain rule, right? I'm kind of curious. I had to change $$e^{\int_0^y Q(t)dt}$$ by putting a negative sign inside the exp. You could also alter it by setting y as the lower limit, of course. Did you hit that snag?

 

[Ald]

Attached is what I did, I haven't rewritten yet. Sometimes ignorance is bliss, I hope I did it right.

 

[Dick]

If ignorance is bliss, I'm experiencing it now. An attachment can take hours to clear approval.