1. The solution of the following partial differential equation $\frac{\partial^2 u}{\partial x^2}=9\frac{\partial^2 u}{\partial y^2}$ is

- sin(3x – y)
- 3x
^{2}+ y^{2} - sin(3x – 3y)
- (3y
^{2}– x^{2})

2. 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 this equation to be classified as parabolic, the value of B^{2} must be

- 1
- 2
- 3
- 4

3. Consider a function f(x,y,z) given by

f(x,y,z) = (x^{2} + y^{2} – 2z^{2})(y^{2} + z^{2})

The partial derivative of this function with respect to x at the point, x = 2, y = 1 and z = 3 is

- 50
- 40
- 30
- 10

4. Consider the following partial differential equation u(x,y) with the constant c > 1

$\frac{\partial u}{\partial y}+c\frac{\partial u}{\partial x}=0$

Solution of this equation is

- u(x,y) = f(x + cy)
- u(x,y) = f(x – cy)
- u(x,y) = f(cx + y)
- u(x,y) = f(cx – y)

5. The type of partial equation

$\frac{\partial^2 P}{\partial x^2}+\frac{\partial^2 P}{\partial y^2}+3\frac{\partial^2 P}{\partial x \partial y}+2\frac{\partial P}{\partial x}-\frac{\partial P}{\partial y}=0$

is

- elliptical
- parabolic
- hyperbolic
- none of these