Solutions to Black Scholes Consider the black scholes equation, 
$$
\frac{\partial V}{\partial t } + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2 } + ( r-q )S\frac{\partial V}{\partial S }-rV =0
$$
How do I show that if $V( S, t)$ is a solution, then $S(\frac{\partial V}{\partial S })$ is also a solution?
I tried substituting $S(\frac{\partial V}{\partial S })$ to the equation and working through the calculations but it doesn't seem to work out.
On a related note, how do we also show that for $ \beta = 1-2(r-q)/\sigma^2$, 
$$
W(S, t) = S^\beta V(\frac{1}{S}, t)
$$
is also a solution?
The relevant partial derivatives are, 
$$
\begin{align}
\frac{\partial W}{\partial S} & = \beta S^{\beta -1 }V- S^{\beta -2}\frac{\partial V}{\partial S}\\
\frac{\partial^2 W}{\partial S^2} & = S^{\beta -4}\frac{\partial^2 V}{\partial S^2} -2S^{\beta -3}\frac{\partial V}{\partial S} + \beta(\beta -1 )S^{\beta-2}V
\end{align}
$$
So the various terms in the PDE are,
$$\begin{align}
(r-q)S\frac{\partial W}{\partial S} & = (r-q) \left[ \beta S^{\beta }V- S^{\beta -1}\frac{\partial V}{\partial S} \right] \\
\frac{1}{2}\sigma^2S^2\frac{\partial^2W}{\partial S^2}& =\frac{1}{2}\sigma^2\left[ S^{\beta -2}\frac{\partial^2 V}{\partial S^2} -2S^{\beta -1}\frac{\partial V}{\partial S} + \beta(\beta -1 )S^{\beta}V \right]
\end{align}
$$
I can see that $ (r-q) \beta S^{\beta }V$ in the top term cancels with $\frac{1}{2}\sigma^2\beta(\beta -1 )S^{\beta}V $ in the bottom term.
So we end up with, 
$$ 
(r-q)S\frac{\partial W}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial^2W}{\partial S^2}=-(r-q) S^{\beta -1}\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2\left[ S^{\beta -2}\frac{\partial^2 V}{\partial S^2} -2S^{\beta -1}\frac{\partial V}{\partial S} \right]
$$
But beyond this I kind of stuck despite trying various manipulations.
Any help will be greatly appreciated!
 A: First note that
$$\frac{\partial}{\partial S}\left(S\frac{\partial V}{\partial S}\right) = \frac{\partial V}{\partial S} + S \frac{\partial^2 V}{\partial S^2}$$
and
$$\frac{\partial^2}{\partial S^2}\left(S\frac{\partial V}{\partial S}\right) = 2\frac{\partial^2 V}{\partial S^2} + S \frac{\partial^3 V}{\partial S^3} \; .$$
Then, take the derivative of the Black-Scholes equation with respect to $S$.
$$\frac{\partial}{\partial T}\frac{\partial V}{\partial S } + \frac{1}{2}\sigma^2 \left(2S\frac{\partial^2 V}{\partial S^2} + S^2 \frac{\partial^3 V}{\partial S^3}\right) + ( r-q )\left(\frac{\partial V}{\partial S } + S\frac{\partial^2 V}{\partial S^2 }\right)-r\frac{\partial V}{\partial S} =0 \; .$$
Using our first two identities, we identify the second and third term
$$\frac{\partial}{\partial T}\frac{\partial V}{\partial S } + \frac{1}{2}\sigma^2 \left(S\frac{\partial^2}{\partial S^2}\left(S\frac{\partial V}{\partial S}\right)\right) + ( r-q )\left(\frac{\partial}{\partial S}\left(S\frac{\partial V}{\partial S}\right)\right)-r\frac{\partial V}{\partial S} =0 \; .$$
Multiplying everything by S, we get
$$\frac{\partial}{\partial T}\left(S\frac{\partial V}{\partial S }\right) + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2}{\partial S^2}\left(S\frac{\partial V}{\partial S}\right) + ( r-q )S\frac{\partial}{\partial S}\left(S\frac{\partial V}{\partial S}\right)-rS\frac{\partial V}{\partial S} =0 \; .$$
A: $$\frac{\partial V}{\partial t } + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2 } + ( r-q )S\frac{\partial V}{\partial S }-rV =0$$
We can regroup
$$\frac{\partial V}{\partial t } + \frac{1}{2}\sigma^2 \left( S\frac{\partial }{\partial S}\right)^2  V + ( r-q - \sigma^2/2 )S\frac{\partial V }{\partial S }  -rV =0.$$
Now observe that $S \frac{\partial}{\partial S}$ commutes with all the coefficients (i.e. terms in front) of $V$ since it commutes itself and doesn't interact with $\frac{\partial}{\partial t}.$
The result is now immediate. 
(see my book Concepts etc section 5.8)
A: Ok for the second part on $W(S,t)=S^\beta V(1/S,t)$, I've figured it out. Just some carelessness on my part when tossing terms around. It goes as follows...
The partial derivatives are,
$$\begin{align}
\frac{\partial W}{\partial S} & = \beta S^{\beta -1 }V- S^{\beta -2}\frac{\partial V}{\partial S}\\
\frac{\partial^2 W}{\partial S^2} & =  \beta \left[ -S^{\beta-3}\frac{\partial V}{\partial S} + (\beta-1)S^{\beta-2}V \right] - \left[ -S^{\beta-4} \frac{\partial^2 V}{\partial S^2} + (\beta-2)S^{\beta -3}\frac{\partial V}{\partial S} \right]
\end{align}$$
Let $x=1/S$ and factor out $S^\beta$, 
$$\begin{align}
\frac{\partial W}{\partial S} & = S^{\beta } \left[ \beta xV -x^2\frac{\partial V}{\partial S} \right]  \\
\frac{\partial^2 W}{\partial S^2}& =  S^{\beta } \left[  \beta \left( -x^3\frac{\partial V}{\partial S} + (\beta-1)x^2V \right)  -\left( -x^4\frac{\partial^2 V}{\partial S^2} +(\beta-2)x^3\frac{\partial V}{\partial S}\right)\right]
\end{align}$$
We exclude $S^{\beta} $ since it's a common factor for all terms. The PDE terms are,
$$\begin{align}
( r-q )S\frac{\partial W}{\partial S} & = ( r-q )x^{-1}\frac{\partial W}{\partial S} \\ & =  ( r-q )\left[ \beta V -x\frac{\partial V}{\partial S} \right]\\
\frac{1}{2}\sigma^2S^2\frac{\partial^2 W}{\partial S^2} &= \frac{1}{2}\sigma^2x^{-2}\frac{\partial^2 W}{\partial S^2}\\ &= \frac{1}{2}\sigma^2 \left[ \beta \left( -x\frac{\partial V}{\partial S} + (\beta-1)V \right)  -\left( -x^2\frac{\partial^2 V}{\partial S^2} +(\beta-2)x\frac{\partial V}{\partial S}\right) \right]
\end{align}$$
As mentioned earlier, the terms involving $V$ cancels, therefore
$$\begin{align}
 ( r-q )S\frac{\partial W}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 W}{\partial S^2} & =  -( r-q )\left[x\frac{\partial V}{\partial S}\right] + \frac{1}{2}\sigma^2\left[(2-2\beta)x\frac{\partial V}{\partial S}+x^2\frac{\partial^2 V}{\partial S^2} \right] \\& =-( r-q )\left[x\frac{\partial V}{\partial S}\right] +2( r-q )\left[x\frac{\partial V}{\partial S}\right] +\frac{1}{2}\sigma^2x^2\frac{\partial^2 V}{\partial S^2}\\&=\frac{1}{2}\sigma^2x^2\frac{\partial^2 V}{\partial S^2}+( r-q )x\frac{\partial V}{\partial S}
\end{align}
$$
Combining with the other two terms we get, 
$$
S^\beta \left[ \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2x^2\frac{\partial^2 V}{\partial S^2}+( r-q )x\frac{\partial V}{\partial S} -rV  \right]=S^\beta.0=0
$$
Therefore we've shown that if $ V(S,t)$ is a solution to the Black Scholes PDE, then $W(S,t)=S^\beta V(1/S,t)$ is also solution. 
