Discussion
If the partial differential equation
\[ \begin{aligned} &(x + 2)\frac{\partial^{2}u}{\partial x^{2}} + 2(x + y)\frac{\partial^{2}u}{\partial x\,\partial y} \\ &+ 2(y - 1)\frac{\partial^{2}u}{\partial y^{2}} - 3y^{2}\frac{\partial u}{\partial y} = 0 \end{aligned} \]
is parabolic on the circle \((x - a)^{2} + (y - b)^{2} = r^{2}\), then the values of \(a\), \(b\) and \(r\) are given by
Answer: B
\(a = -1,\ b = 2,\ r = 1\).
A second-order equation \(Au_{xx} + 2Bu_{xy} + Cu_{yy} + \ldots = 0\) is parabolic where \(B^{2} - AC = 0\). Here \(A = x + 2\), \(2B = 2(x + y)\) so \(B = x + y\), and \(C = 2(y - 1)\):
\[ \begin{aligned} (x + y)^{2} - 2(x + 2)(y - 1) &= 0 \\ x^{2} + y^{2} + 2x - 4y + 4 &= 0 \\ (x + 1)^{2} + (y - 2)^{2} &= 1 \end{aligned} \]
The circle has centre \((-1, 2)\) and radius 1. The other options flip the sign of a or b, which is what happens if the completed squares are read as \((x - 1)\) or \((y + 2)\).
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