Is it possible for a quadratic surface to have variables and coefficients that are functions? For example, an equation such as
$$A(T)x(T)^2 + B(T)y(T)^2 + C(T)z(T)^2 + D(T)x(T)y(T)\;...$$ where both the variables and the coefficients are functions of $T$.
The reason I ask is because I have a model with an equilibrium at at given temperature $T$, $$H^*(T) = \frac{z\mu_I(T)\left(1+\frac{\mu_U(T)}{\theta(T)}\right)}{\phi(\lambda-z\left(1+\frac{\mu_U(T)}{\theta(T)}\right)}$$
whose derivative, when set equal to zero, ends up being of the form in the first equation shown above. It looks just like a quadratic surface except that the variables $x,y,z$ are functions of temperature, and the coefficients $A,B,C$ contain the derivatives of $x,y,z$. (Here $x = \theta, y = \mu_U$ and $z = \mu_I$).
Namely,
$$\theta(T)^2\left(z\mu_I'(T)-\lambda\mu_I'(T)\right) + \mu_U(T)^2\left(z\mu_I'(T)\right) + \theta(T)\mu_U(T)\left(2z\mu_I'(T)-\mu_I'(T)\right) + \mu_U(T)\mu_I(T)\left(\theta'(T)\lambda\right) - \theta(T)\mu_I(T)\left(\lambda\mu_U'(T)\right) = 0$$
I thought that it would be useful to put this into normal form and plot how the surface changed as temperature changed (or at least determine what type of surface it is), but I couldn't find examples of quadratic surfaces that contain functions like this and I don't want to go down this road if it's nonsense. Any help would be greatly appreciated.