Partial Differential Equations MCQ

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

1. sin(3x – y)
2. 3x² + y²
3. sin(3x – 3y)
4. (3y² – x²)

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 B² must be

1. 1
2. 2
3. 3
4. 4

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

f(x,y,z) = (x² + y² – 2z²)(y² + z²)

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

1. 50
2. 40
3. 30
4. 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

1. u(x,y) = f(x + cy)
2. u(x,y) = f(x – cy)
3. u(x,y) = f(cx + y)
4. 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

1. elliptical
2. parabolic
3. hyperbolic
4. none of these