Is an analytic solution possible for this problem: find an expression for maximum of p subject to constraints f=0, g=0, h=0? The feasible range is: T1, T2, R, Rthhot, Rthcold, and Z>0.

I know how to solve it using either FindMaximum or Langrage multiplier if I assign numerical values to T1, T2, R, Rthhot, Rthcold, and Z. I'd like to know if it's possible to find an analytic solution if I leave T1, T2, R, Rthhot, Rthcold and Z as parameters.

p[X_, Y_, Rload_, Rth_, S_] := S^2*(X - Y)^2*Rload/(R + Rload)^2;

f[X_, Y_, Rload_, Rth_, S_] := -(T1 - X)/Rthhot + (X - Y)/Rth + S^2*X*(X - Y)/(R + Rload) - 
0.5*S^2*Rload*(X - Y)^2/(R + Rload)^2;

g[X_, Y_, Rload_, Rth_, S_] := (Y - T2)/Rthcold - (X - Y)/Rth - S^2*Y*(X - Y)/(R + Rload) - 
0.5*S^2*Rload*(X - Y)^2/(R + Rload)^2;

h[X_, Y_, Rload_, Rth_, S_] := S^2*Rth/R - Z;