Show that the equation 𝑢𝑥𝑥 + 𝑥𝑢𝑦𝑦 + 𝑢𝑦 = 0 is elliptic for 𝑥 > 0 and hyperbolicfor 𝑥 < 0.

The classification of a partial differential equation (PDE) as elliptic, parabolic, or hyperbolic depends on the discriminant of the equation, which is given by B^2 - 4AC for a second-order PDE of the form Au_xx + 2Bu_xy + Cu_yy + ... = 0.

In the given equation, u_xx + xu_yy + u_y = 0, we have:

A = 1 (coefficient of u_xx) B = 0 (there is no term with u_xy) C = x (coefficient of u_yy)

So, the discriminant is B^2 - 4AC = 0^2 - 41x = -4x.

For x > 0, the discriminant is negative, which means the equation is elliptic.

For x < 0, the discriminant is positive, which means the equation is hyperbolic.

Therefore, the given equation is elliptic for x > 0 and hyperbolic for x < 0.