= Solution
Multiply the <Sturm-Liouville form> for $y_n$ by $y_m$ and subtract the equation for $y_m$ multiplied by $y_n$. Integration gives
$$
(n^2-m^2)\int_{-1}^1w\,y_ny_m\,dx
=\left[p(y_ny_m'-y_my_n')\right]_{-1}^1=0.
$$
The boundary expression is zero because $p\to0$ and the functions and first <derivatives> remain finite. Since $n,m$ are distinct positive integers, this proves
$$
\boxed{\int_{-1}^1y_ny_m\sqrt{1-x^2}\,dx=0.}
$$
For the <derivatives>, apply <integration by parts> once to the same <Sturm-Liouville form>:
$$
\int_{-1}^1p\,y_n'y_m'\,dx
=\left[py_n'y_m\right]_{-1}^1-\int_{-1}^1(py_n')'y_m\,dx
=(n^2-1)\int_{-1}^1w\,y_ny_m\,dx=0.
$$
Thus the new weight is
$$
\boxed{w_1(x)=(1-x^2)^{3/2}.}
$$
This is <derivative orthogonality when the Sturm-Liouville potential vanishes>. For $n=1$ the <derivative> is identically zero, which is consistent with the identity.
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