Using the transmutation theorem of Leibniz

[burt]

I was asked the following:
>Given the curve $y^q=x^p (q>p>0)$ show using the transmutation theorem that $\int^{x_0}_0ydx=\frac{qx_0y_0}{p+q}$
Note that from $y^q=x^p$ it follows that $q \frac{dy}{y}=p\frac{dx}{x}$. Therefore, $z=y-x\frac{dy}{dx}=(\frac{q-p}{q})y$.

I'm not really sure how show this. Here is my work:
The transmutation theorem states: $\int^{x_0}_0ydx=\frac12(x_0y_0+\int^{x_0}_0z dx)$

We know z in this case and so can write it out like this: $\int^{x_0}_0ydx=\frac12(x_0y_0+\int^{x_0}_0y-x\frac{dy}{dx} dx)=\frac12(x_0y_0+\int^{x_0}_0y-x dx)$
=$\frac12(x_0y_0+yx_0-\frac{x_0^2}{2})$

Am I going about this right? How can I proceed from here?

 

[burt]

I was asked the following:
>Given the curve $y^q=x^p (q>p>0)$ show using the transmutation theorem that $\int^{x_0}_0ydx=\frac{qx_0y_0}{p+q}$
Note that from $y^q=x^p$ it follows that $q \frac{dy}{y}=p\frac{dx}{x}$. Therefore, $z=y-x\frac{dy}{dx}=(\frac{q-p}{q})y$.

I'm not really sure how show this. Here is my work:
The transmutation theorem states: $\int^{x_0}_0ydx=\frac12(x_0y_0+\int^{x_0}_0z dx)$

We know z in this case and so can write it out like this: $\int^{x_0}_0ydx=\frac12(x_0y_0+\int^{x_0}_0y-x\frac{dy}{dx} dx)=\frac12(x_0y_0+\int^{x_0}_0y-x dx)$
=$\frac12(x_0y_0+yx_0-\frac{x_0^2}{2})$

Am I going about this right? How can I proceed from here?

I keep moving things around - and using the different values for z I was able to solve for $x_0$. I got $x_0=\frac{2py}{q}$. But, I don't see how this helps me.

 

[HallsofIvy]

The "transmutation theorem" doesn't make sense to me because the undefined "z" mysteriously pops up. You give a value for z in this particular case without saying how it is defined.

Without the "transmutation theorem", I would say, since $\displaystyle x^p= y^q$, that $\displaystyle y= x^{p/q}$. Then $\displaystyle \int_0^{x_0} x^{p/q}dx= \frac{1}{\frac{p}{q}+ 1}\left[x^{\frac{p}{q}+1}\right]_0^{x_0}$
$\displaystyle = \frac{q}{p+q}x_0(x_0^{p/q})= \frac{q}{p+q}x_0y_0$
(assuming that "$\displaystyle y_0$" is the value of y when $\displaystyle x= x_0$ which you did not actually say).

 

[burt]

You give a value for z in this particular case without saying how it is defined.

The value of z was given to me in the original problem. It is part of what is contributing to my confusion.