Discussion
The equation \(3u_{xx} - x^{2}u_{y} + 5u = 0\) transforms under the change of variables to polar coordinates is :
Answer: D
The equation remains a parabolic equation.
The type of a second-order linear PDE \(Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots = 0\) is decided by the sign of \(B^{2} - AC\). Here \(A = 3\) and \(B = C = 0\) (\(u_{y}\) is first order), so \(B^{2} - AC = 0\): the equation is parabolic.
A smooth, invertible change of variables such as polar coordinates multiplies the discriminant by the square of the Jacobian, so its sign, and hence the type, does not change. The equation is also linear in \(u\), so it does not become non-linear, and it cannot become hyperbolic or elliptic.
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