High School First Order Non-Linear ODE (what method to use?)

[Apothem]

Hi,

The problem is to solve:
dy/dx = −[2x + ln(y)]*(y/x)

Attempt:
I have tried to see if it is exact, I found it not to be, I can't easily find a function to multiply by to make it exact either (unless I am missing something obvious). It clearly isn't seperable, nor is it homogenous (I know although some non-homogenous ODEs can be solved using the homogenous method I don't think this one can be), it also isn't a linear equation so we can't use integrating factors.

I'm looking for pointers in the right direction/which method to use.

Thanks for your time

 

[jedishrfu]

Have you tried guessing a solution?

Something like $a^x$ or something that works well with the natural log function.

 

[jedishrfu]

Did you figure it out?

 

[Ravikant Rajan]

I think you are going the right way; checking the exact form, homogeneity etc.
Here since there is a logarithmic function, I recommend substituting y=e^t where t is parameter and then be solved by finding integrating factor.

 

[John Park]

Using y = exp(qx), where q is a function of x , I got it to separate, giving
q = (A/x²) - 1, where A is a constant of integration.
It seems to check.