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.