Question

The differential equation \[ \frac{\partial^2u}{\partial x^2} +4\frac{\partial^2u}{\partial x\partial y} +4\frac{\partial^2u}{\partial y^2} -\frac{\partial u}{\partial x} -2\frac{\partial u}{\partial y} =0 \] is classified as

• A. Parabolic
• B. Elliptic
• C. Hyperbolic
• D. Both Parabolic & Elliptic

Hint:

For \[ Au_{xx}+2Bu_{xy}+Cu_{yy}=0, \] \[ \boxed{ \begin{aligned} B^2-AC>0 &\rightarrow \text{Hyperbolic} B^2-AC=0 &\rightarrow \text{Parabolic} B^2-AC<0 &\rightarrow \text{Elliptic} \end{aligned} } \]

Answer 1

The Correct Option is A

Step 1: Compare with the standard second-order PDE. A second-order PDE is written as \[ A\frac{\partial^2u}{\partial x^2} +2B\frac{\partial^2u}{\partial x\partial y} +C\frac{\partial^2u}{\partial y^2} +\cdots=0. \] Here, \[ A=1,\qquad 2B=4\Rightarrow B=2,\qquad C=4. \]

Step 2: Evaluate the discriminant. The discriminant is \[ B^2-AC =2^2-(1)(4) =4-4 =0. \] Since \[ \boxed{B^2-AC=0,} \] the equation is \[ \boxed{\text{Parabolic}.} \] Therefore, \[ \boxed{(A)} \] is the correct answer.