First Order Differential (Just identify)

[Vathral]

Not sure if I should have put this here or in the homework section but I am simply asking for direction on these problems.

I have 5 problems to work on and my biggest problem is identifying what method to use to find the answer. I don't need help answering because it's something I'd like to do myself but have some trouble with as I learn.


Don't know how to use the neat forum stuff yet but will learn as I become part of the community slowly.

Problems to work on...
http://img261.imageshack.us/img261/9066/matwk8.jpg

Those are the problems and so far I can see...

#1 ?
#2 is a separable equation correct?
#3 ?
#4 is substitution and then exact?

Already marked that one exact since I did it already.
Can't tell which one would follow Bernoulli equation, integrating factor, linear first order.


Thank you to anyone who would kindly reply :x


Edit: Took a look at the rules again and see that this really should belong in the homework section. I'm very sorry, I'll ask an admin/mod to move it.

 

[tiny-tim]

Welcome to PF!

Hi Vathral! Welcome to PF!

For #1, try the obvious substitution.

 

[Vathral]

$$x \frac{dy}{dx} - y = \frac{x^3}{y} e^y_x$$

So
$$v = \frac{y}{x}$$

$$y = vx$$

$$\frac{1}{y} = \frac{x}{y}$$

$$\frac{dy}{dx} = v+x \frac{dv}{dx}$$



$$x [v + x\frac{dv}{dx}] - vx = \frac{x^2}{v} e^v$$

$$xv + x^2 \frac{dv}{dx} - vx = \frac{x^2}{v} e^v$$

$$x^2 \frac{dv}{dx} = \frac{x^2}{v} e^v$$

$$\frac{dv}{dx} = \frac{e^v}{v}$$

$$\frac{vdv}{e^v} = dx$$

$$\frac{-v-1}{e^v} = x + c$$

$$\frac{-\frac{y}{x}- 1)}{e^y_x} = x + c$$

$$- \frac{y}{x} - 1 = (x + c) e^y_x$$

$$-y - 1 * x^2 e^y_x + cxe^y_x$$

$$y = x^2 e^y_x+cxe^y_x$$

This is for #1 of course. Anyone see anything wrong with the final answer? Thank you for the identifying, would like to take these questions an extra step now.

 

[tiny-tim]

$$- \frac{y}{x} - 1 = (x + c) e^y_x$$

$$-y - 1 * x^2 e^y_x + cxe^y_x$$

$$y = x^2 e^y_x+cxe^y_x$$

Hi Vathral!

Fine up until that line …

but what happened to the - 1 on the left?

For question 3, the only problem is the mixed terms (mixed x and y) …

so start by ignoring the 8x²dx, and try and turn the other two into an exact differential … and then put the 8x²dx back in, of course.