How to Simplify the Solution of y' = 6y ln(y)/x?

[Saladsamurai]

Solve:

$$y' = 6\frac{y\ln y}{x}$$

After separation and integration, I got

$$\ln[\ln y] = 6\ln x + c_1$$

$$\Rightarrow \ln y = e^{\ln x^6 + c_1}$$

I am not sure how to get this into an explicit form for y, without it getting nasty. I know that there is usually a trick to make it look cleaner.

Any thoughts?

 

[Saladsamurai]

On a whim, I let $c_1 = \ln c_2$ and good things happen! I get,

$$y = e^{\frac{x^6}{c_2}}$$

 

[HallsofIvy]

Acutally, if you let $c_1= ln(c_2)$ then you would have $ln(x^6)+ c_1= ln(x^6)+ ln(c_2)= ln(c_2x^6)$ so that
$$ln(y)= e^{ln(c_2x^6)}= c_2x^6$$
and
$$y= e^{c_2x^6}$$.

That is, $c_2$ is multiplying $x^6$, not dividing it. But since it is simply an arbitrary constant, it really does not matter.

 

[Saladsamurai]

Oops! It was a summation of logs! Not a difference ... Nice catch Halls!