Help with an (I think) homogeneous DE.

[1MileCrash]

Homework Statement

$y' = \frac{2xy + y^{2} + 1}{y(2+3y)}$

Homework Equations

The Attempt at a Solution

First I tried making a substitution in the case that it is homogeneous, but it didn't make the equation separable. It's not linear, it's not exact, and not separable.

Does it become exact when multiplying by some function?

I just need a little guidance for what method I should use to solve.

After making a substitution y = vx,

$v + xv' = \frac{2x^{2}v + v^{2}x^{2} + 1}{vx(2+3vx)}$

This doesn't seem to simplify into anything separablem, after doing some algebra. Any ideas?

 

[LCKurtz]

Homework Statement

$y' = \frac{2xy + y^{2} + 1}{y(2+3y)}$First I tried making a substitution in the case that it is homogeneous,

but it didn't make the equation separable.

You don't want to waste time trying the $y=vx$ substitution on the offhand chance it might be homogeneous. You write it as $Mdx + Ndy = 0$ and check whether or not $M$ and $N$ are homogeneous of the same degree. In this case neither $M$ nor $N$ are homogeneous of any degree, much less the same degree.

Other than that observation, I agree with what you say about the equation. Unfortunately, I don't have any helpful suggestions on what to do with this one. I presume you know it is not a given that a random DE like this admits an easy solution. Where did you get this problem?

 

[1MileCrash]

It's just in the problem set of my textbook, after covering a few methods. I'm pretty sure I wrote it down correctly.

Thanks

 

[1MileCrash]

Nope, copied the numerator from one and denomenator from that other.

Thanks for the help. I'll be sure to apply what you said about testing for homogenous equations.