Proof of Integration Factor for diff eq

In summary: Please type out the important steps in a readable form.In summary, the conversation revolves around a first order differential equation problem and finding an integrating factor. The participants discuss proving exactness and using the formula for mu(y). The conversation also includes a clarification of the term 'wrt' and a question about whether the method worked. The conversation ends with one participant sharing their solution and requesting the other participant to type out the important steps in a readable form.
  • #1
Ald
25
1

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

 
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  • #2
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.
 
  • #3
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
 
  • #4
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.
 
  • #5
I'm sorry what does 'wrt' mean?
 
  • #6
wrt='with respect to'. What's the derivative of mu(y) with respect to y?
 
  • #7
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
 
  • #8
Thanks Dick for getting me in right direction, I think I have it proved.
Thanks again
 
  • #9
Ald said:
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 [tex]e^{\int_0^y Q(t)dt}[/tex] 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?
 
  • #10
Attached is what I did, I haven't rewritten yet. Sometimes ignorance is bliss, I hope I did it right.
 

Attachments

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  • #11
If ignorance is bliss, I'm experiencing it now. An attachment can take hours to clear approval.
 

FAQ: Proof of Integration Factor for diff eq

What is a proof of integration factor for differential equations?

A proof of integration factor for differential equations is a mathematical technique used to solve certain types of differential equations. It involves finding a function, known as an integration factor, that can be multiplied with the original differential equation to make it easier to solve.

Why is the proof of integration factor important?

The proof of integration factor is important because it allows us to solve certain types of differential equations that may be too difficult or impossible to solve otherwise. It also provides a systematic approach to solving these equations, making the process more efficient.

How do you find the integration factor for a differential equation?

The integration factor is typically found by multiplying both sides of the original differential equation by an appropriate function, such as the inverse of the coefficient of the highest derivative term. This will reduce the differential equation to one that is easier to solve.

What types of differential equations require the use of an integration factor?

The most common types of differential equations that require the use of an integration factor are those that are not exact, meaning that their solutions do not satisfy the equality of mixed partial derivatives. These include equations with non-constant coefficients and equations that are not in the form of separable variables.

Are there any limitations to using the proof of integration factor for differential equations?

While the proof of integration factor is a useful technique for solving certain types of differential equations, it may not work for all equations. It is important to carefully analyze the equation to determine if this method is appropriate before attempting to use it. Additionally, the integration factor may not always be easy to find, and the resulting solution may still require further manipulation to obtain the final answer.

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