Classify the following partial differential equations:(i) 𝜕2𝑢𝜕𝑥2 + 3 𝜕2𝑢𝜕𝑦2 = 𝜕𝑢

The given partial differential equation is:

𝜕²𝑢/𝜕𝑥² + 3 𝜕²𝑢/𝜕𝑦² = 𝜕𝑢

To classify this PDE, we need to look at the highest order derivatives. In this case, we have second order derivatives with respect to x and y.

The general form of a second order PDE is:

A(𝜕²𝑢/𝜕𝑥²) + B(𝜕²𝑢/𝜕𝑦²) + C(𝜕𝑢/𝜕𝑥) + D(𝜕𝑢/𝜕𝑦) + E𝑢 = F

Where A, B, C, D, E are coefficients which can be functions of x and y.

The classification of the PDE depends on the discriminant B² - 4AC.

In our case, A = 1, B = 3, C = 0, D = 0, E = 0. So, the discriminant is B² - 4AC = 3² - 410 = 9.

Since the discriminant is positive, the given PDE is an elliptic type.