1. ## homogeneous equation

I keep getting stuck on how to separate the V once I substitute.

2. Well, show us your work so far.

3. Okay, my first step

$\displaystyle (xdv+vdx) = (vx(1+lnvx - lnx)/x) dx$
For this just divided x out
$\displaystyle (xdv+vdx) = (v(1+lnvx - lnx) dx$
2. I separated (lnvx) = lnv + lnx, in order to eliminate lnx then I multiplied V
$\displaystyle (xdv+vdx) = (v+ vlnv ) dx$
3. I brought vdx over and subtracted V
$\displaystyle (xdv) = (vlnv ) dx$
$\displaystyle (1/vlnv(dv) = (1/x ) dx$
$\displaystyle \int [(1/vln(v)dv] = \int[ (1/x )] dx$
$\displaystyle (1/4)v^2 (2ln(v) -1) = ln x$
4. multiplied everything by e
$\displaystyle 1/4v^2(v^2-1) = x$
This as far as I could go( not even sure if it's right) when I substituted v = y/x, I can't solve for y.

4. I like everything up to, but not including, your integrations. If you're integrating

$\displaystyle \displaystyle\int\frac{dv}{v\,\ln(v)},$

then let $\displaystyle u=\ln(v),\;du=\dfrac{dv}{v},$ and the integral becomes

$\displaystyle \displaystyle\int\frac{du}{u}=\ln|u|+C=\ln|\ln(v)| +C.$

Carry that correction through. What do you get?

5. $\displaystyle v = lnx$
$\displaystyle y/x = lnx$
$\displaystyle y=xlnx$
I believe this is the right answer.

Sometimes I mess up on my integrals...

6. No, no. Logarithms, when iteratively applied, do NOT cancel out. You have this:

$\displaystyle \ln|\ln(v)|=\ln|x|+C.$ Exponentiate both sides and relabel the constant:

$\displaystyle \ln(v)=Cx,$ and hence

$\displaystyle \ln(y/x)=Cx.$

Can you continue?

7. hmm... I truly need to review logs

$\displaystyle e^(ln(y/x)) = e^(cx)$
$\displaystyle y/x=Ce^x$
$\displaystyle y= xCe^x$

Thx once again.... I believe I got it right. The only other way that I could is by splitting Lny/lnx = Lny - lnx, if I did this way the answer would be the same.

8. You can know that you are right. Differentiate this solution and see if it satisfies the original DE. I would recommend doing this for every single DE you ever solve. It's a good, straight-forward exercise (differentiation is a lot easier than integration), and it gives you confidence that you got the right answer.

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# integration of 1/vlnv

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