Pricing equation for a call on the squared asset

A stock $S$ follows geometric Brownian motion. What partial differential equation does the price of a call on $S^2$ satisfy?

The ordinary Black-Scholes PDE in $S$: $V_t+\tfrac12\sigma^2S^2V_{SS}+rSV_S-rV=0$, with $V(S,T)=(S^2-K)^+$. Any tradeable derivative on $S$ satisfies it, since the hedge is in $S$. In the variable $Y=S^2$, which has risk-neutral drift $(2r+\sigma^2)Y$ and volatility $2\sigma$, it reads $V_t+2\sigma^2Y^2V_{YY}+(2r+\sigma^2)YV_Y-rV=0$.