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)
Answer. a

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
Answer. c

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
Answer. b

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)
Answer. b

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


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