Integrating Factors Found by Inspection

[heykyou]

I need some explanation regarding a solution found in our textbook.

The example starts with this DE:

(1) y(x³ - y)dx - x(x³ + y)dy = 0.

By regrouping, exact DEs will be found and the equation can be rewritten to:

(2) x³ d($$\frac{x}{y}$$) - $$\frac{d(xy)}{y}$$ = 0.

The integrating factor is found to be: x^-2y^-1. Multiplying the equation by the IF results to:

(3) $$\frac{x}{y}$$ d($$\frac{x}{y}$$) - $$\frac{d(xy)}{x^{2}y^{2}}$$

Now this is the part that I don't understand. The text continues after the equation above, "of which a set of solutions is given by"

(4) $$\frac{1}{2}$$ ($$\frac{x}{y}$$)² + $$\frac{1}{xy}$$ = $$\frac{c}{2}$$.

Just how did (3) turn into (4)?

I think it's a very simple question but I'm really lost. I'm so confused with anything that has to do with differential equations. :( Thanks in advance for the help.

 

[CompuChip]

I suppose they just integrated equation 3.
If we call u = x / y and v = x y, then putting an integral sign in front of (3) reads
$$\int u \, du - \int \frac{dv}{v^2} = 0.$$
Both integrals are elementary, giving
[tex]\tfrac12 u^2 + \frac{1}{v} + C[/itex].<br /> We can now plug in the definitions of u and v again, and bring C to the other side, giving<br /> $$\frac12 \left( \frac{x}{y} \right)^2 + \frac{1}{xy} = \frac{c}{2}$$<br /> where c = -2C is just another constant (i.e. determining C from boundary conditions is equivalent to determining c).<br /> <br /> If this means more to you: basically, they just did a change of variables.[/tex]

 

[heykyou]

Thanks!

Yeah, I knew there was some integration going on, but I didn't know how to integrate the x's and y's since they were together. I get it now. The exact DE's were absorbed into the du and dv. Stupid. :( Thanks so much for taking the time to explain it step by step. :)

I want to ask another question, though, if it's all right. (Maybe I should change the topic title to something more general--because I have so many questions regarding DE's...)

I'm having problems with transformations of homogeneous equations as well. I get to a solution, but it's not the solution in the book. For example,

(16x + 5y)dx + (3x + y)dy = 0

I used the transformation y = vx, y' = x + xv'. Then I divided by $$\frac{1}{x^{2}}$$, put the fractions together and ended up with:

$$\frac{dx}{x}$$ = $$\frac{(3 + v)dv}{(v + 4)^{2}}$$

Integrating using partial fractions, the result is:

-ln|x| = ln|v + 4| - $$\frac{1}{v + 4}$$ + c

I plugged back v = y/x, solved for c (x = 1; y = -3), and so the answer is:

-y - 3x = (y + 4x) ln(y + 4x)

But the book says the answer is: y + 3x = (y + 4x) ln(y + 4x).

My answers often have incorrect signs. Was there anything wrong in the process I did? Or would they be equal in the end? I double-checked it, and still ended up with the same answer. And I have this problem for most of transformations of homogeneous equations.

Thanks again in advance. :)

 

[CompuChip]

I'm not sure where all those minus signs came from, but
$$\int \frac{dx}{x} = \ln|x|$$
and
$$\ln|v + 4| + \frac{1}{v + 4}$$
according to Mathematica.
Perhaps you have introduced some faulty signs in the partial fractions?

Maybe an easier solution would be to write
$$\frac{3 + v}{(v + 4)^2} = \frac{(4 + v) - 1}{(v +4)^2} = \frac{1}{v + 4} - \frac{1}{(v + 4)^2}.$$

 

[heykyou]

Thanks so much, CompuChip. :)

I worked on that equation again. BTW, sorry, I made a typo--there is supposed to be a negative sign before dx/x, so the integration -ln|x| is correct. My mistake was in the integration by partial fraction. My resulting integral was separated by a negative sign when it should have been a plus. That fixed everything! Thanks! :)

I worked on the other homogeneous equations over the week as well. Most errors were due to integrating mistakes. :( Now I feel embarrassed. :lol:

I do need help with other equations again, however. Sorry in advance for so much trouble. I've been practicing on a lot of DE's, so I have a rather big unanswered list. Here goes:

$$(1 - xy)^{-2}dx + [y^{2} + x^{2}(1 - xy)^{-2}]dy = 0$$

The above equation is not separable, not exact, not linear, and not homogeneous. Or is it? I'm out of ideas for this one. I did try to multiply it by (1 - xy)², but it didn't seem to help.

The next two are supposed to be answered by looking for a common exact DE by regrouping the terms, but any pattern is really difficult to find:

$$y(x^{2} + y)dx + x(x^{2} - 2y)dy = 0$$
$$y(3x^{3} - x + y)dx + x^{2}(1 - x^{2})dy = 0$$

And the topic that stumps me most: Bernoulli equations! I haven't been able to solve even one equation. :( It'd be too much to post all equations here, so I'll just post the first one. Hopefully, after I know how to solve this one, I'll be able to get the drift of it.

$$\frac{dy}{dx} = sin(x + y)$$

No matter how I look at it, it just doesn't seem to be a Bernoulli equation. It's nowhere near the correct form. :(

All right, sorry, sorry, sorry for so many questions and thanks, thanks, thanks. :D Any help with any equation is very much appreciated!Edit

I have another question. *sheepish* I'm wondering what the general approach to answering DE's is. In what order should you check the form of the DE? Like, I know the first you have to check for is separability. Then? Exactness? Linearity? Homogeneity? Thanks in advance!