Question

Consider the following partial differential equation

\(3\frac{{{\partial ^2}\phi }}{{\partial {x^2}}} + B\frac{{{\partial ^2}\phi }}{{\partial x\partial y}} + 3\frac{{{\partial ^2}\phi }}{{\partial {y^2}}} + 4\phi = 0\)

For the equation to be classified as parabolic, the value of B² must be _______.

Answer 1

\(3\frac{{{\partial ^2}\phi }}{{\partial {x^2}}} + B\frac{{{\partial ^2}\phi }}{{\partial x\partial y}} + 3\frac{{{\partial ^2}\phi }}{{\partial {y^2}}} + 4\phi = 0\)

Compare above equation with \(A\frac{{{\partial ^2}\phi }}{{\partial {x^2}}} + B\frac{{{\partial ^2}\phi }}{{\partial x\partial y}} + C\frac{{{\partial ^2}\phi }}{{\partial {y^2}}} + D\phi = 0\)

A = 3      B = ?      C = 3

For equation to be parabolic

B² – 4AC = 0

B² – 4 × 3 × 3 =

B² = 36