Topic: Electronics Engineering
For $-\infty < x < \infty$ and $-1 < y < 1$, the partial differential equation $x^{2}\frac{\partial^{2}u}{\partial x^{2}}+(1-y^{2})\frac{\partial^{2}u}{\partial y^{2}}=0$ is
- A. Parabolic
- B. Hyperbolic
- C. Elliptic
- D. Nature cannot be predicated
Answer: C. Elliptic
Explanation: Comparing with $A\frac{\partial^{2}u}{\partial x^{2}} + B\frac{\partial^{2}u}{\partial x\partial y} + C\frac{\partial^{2}u}{\partial y^{2}} = 0$, we get $A = x^{2}$, $B = 0$, $C = 1-y^{2}$. Now calculate $B^{2}-4AC = 0 - 4(x^{2})(1-y^{2})$. For $-1 < y < 1$, $(1-y^{2}) > 0$. Hence $B^{2}-4AC < 0$ (for $x \neq 0$), which is the condition for an Elliptic partial differential equation.