Why Is My Integrating Factor Not Solving the Exact ODE?

[ozone]

I'm not sure where I'm going wrong on this one so I hoped that I could find some help

we begin with
$(x^2 + y^2 + 5) dx - (y+xy) dy$

taking both partial derivitives I found that

$2y (dy) =/ -y(dx)$

Next I went to find my factor of integration using $e^(My - Nx / N) dx)$This got me $((1+x)^-3)$

which i then simplified to $(1 + 1/x^3)$Then i multiplied our I.F. through the original M and N, but the problem still did not come out to be equal

our new partial derivitives of m and n are:

$((2y/x^3)(dy) =/ ((3y/x^4) + (2y/x^3) - (y))(dx))$Sorry I couldn't figure out how to display notequal with itex.. anyways thanks in advance for any help

 

[HallsofIvy]

I'm not sure where I'm going wrong on this one so I hoped that I could find some help

we begin with
$(x^2 + y^2 + 5) dx - (y+xy) dy$

taking both partial derivitives I found that

$2y (dy) =/ -y(dx)$

Next I went to find my factor of integration using $e^(My - Nx / N) dx)$


This got me $((1+x)^-3)$

which i then simplified to $(1 + 1/x^3)$

Well, that's a problem! $(1+x)^{-3}$ is NOT equal to
$$1+ \frac{1}{x^3}$$
It is, rather,
$$\frac{1}{(1+ x)^3}$$

Then i multiplied our I.F. through the original M and N, but the problem still did not come out to be equal

our new partial derivitives of m and n are:

$((2y/x^3)(dy) =/ ((3y/x^4) + (2y/x^3) - (y))(dx))$


Sorry I couldn't figure out how to display notequal with itex.. anyways thanks in advance for any help

 

[ozone]

Sooo wolfram's mistaken??

http://www.wolframalpha.com/input/?i=distribute+(1+x)^-3

 

[HallsofIvy]

What do you think "distribute" means there?

 

[blue_raver22]

Hello,

It seems like you are on the right track with your approach to solving the exact ODE problem. However, it is important to double check your calculations and make sure that you are using the correct formulas and methods. Here are a few things to keep in mind:

1. When taking the partial derivatives, make sure to use the correct notation. It should be d/dx for the derivative with respect to x and d/dy for the derivative with respect to y. Also, make sure to treat the other variable as a constant when taking the derivative.

2. When finding the integrating factor, the formula should be e^(My - Nx) instead of e^(My - Nx / N). Make sure to use the correct formula to avoid any errors.

3. When multiplying the integrating factor through the original ODE, make sure to distribute it correctly. It should be (1+x)^-3 * (x^2 + y^2 + 5)dx - (1+x)^-3 * (y+xy)dy.

4. After multiplying the integrating factor, you should end up with the exact ODE. If it is not equal, then there may be a mistake in your calculations. Double check your work and make sure that you are using the correct formulas and methods.

I hope this helps and good luck with solving the exact ODE problem! Remember to always double check your work and ask for help if you are unsure.