Partial Differential Equations Questions and Answers – First Order Linear PDE

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This set of Fourier Analysis and Partial Differential Equations Multiple Choice Questions & Answers (MCQs) focuses on “First Order Linear PDE”.

1. First order partial differential equations arise in the calculus of variations.
a) True
b) False

Explanation: The calculus of variations is a type of analysis in the field of mathematics (branch of calculus) which is used to find maxima and minima of definite integrals.

2. The symbol used for partial derivatives, ∂, was first used in mathematics by Marquis de Condorcet.
a) True
b) False

Explanation: Partial derivatives are indicated by the symbol ∂. This was first used in mathematics by Marquis de Condorcet who used it for partial differences.

3. What is the order of the equation, $$xy^3(\frac{∂y}{∂x})^2+yx^2+\frac{∂y}{∂x}=0$$?
a) Third Order
b) Second Order
c) First Order
d) Zero Order

Explanation: The equation having only first derivative, i.e., $$\frac{∂y}{∂x}$$ are said to be first order differential equation. Since the given equation satisfies this condition, it is of first order.
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4. In the equation, y= x2+c,c is known as the parameter and x and y are known as the main variables.
a) True
b) False

Explanation: Given: y= x2+c, where c is known as an arbitrary constant. It is also referred to as the parameter to differentiate it from the main variables x and y.

5. Which of the following is one of the criterions for linearity of an equation?
a) The dependent variable and its derivatives should be of second order
b) The dependent variable and its derivatives should not be of same order
c) Each coefficient does not depend on the independent variable
d) Each coefficient depends only on the independent variable

Explanation: The two criterions for linearity of an equation are:

• The dependent variable y and its derivatives are of first degree.
• Each coefficient depends only on the independent variable

6. Which of the following is a type of Iterative method of solving non-linear equations?
a) Graphical method
b) Interpolation method
c) Trial and Error methods
d) Direct Analytical methods

Explanation: There are 2 types of Iterative methods, (i) Interpolation methods (or Bracketing methods) and (ii) Extrapolation methods (or Open-end methods).

7. Which of the following is an example for first order linear partial differential equation?
a) Lagrange’s Partial Differential Equation
b) Clairaut’s Partial Differential Equation
c) One-dimensional Wave Equation
d) One-dimensional Heat Equation

Explanation: Equations of the form Pp + Qq = R , where P, Q and R are functions of x, y, z, are known as Lagrange’s linear equation.

8. What is the nature of Lagrange’s linear partial differential equation?
a) First-order, Third-degree
b) Second-order, First-degree
c) First-order, Second-degree
d) First-order, First-degree

Explanation: Lagrange’s linear equation contains only the first-order partial derivatives which appear only with first power; hence the equation is of first-order and first-degree.

9. Find the general solution of the linear partial differential equation, yzp+zxq=xy.
a) φ(x2-y2 – z2 )=0
b) φ(x2-y2, y2-z2 )=0
c) φ(x2-y2, y2-x2 )=0
d) φ(x2-z2, z2-x2 )=0

Explanation: Given: yzp+zxq=xy
Here, the subsidiary equations are, $$\frac{dx}{yz}=\frac{dy}{zx}=\frac{dz}{xy}$$
From the first two and last two terms, we get, respectively,
$$\frac{dx}{yz}=\frac{dy}{zx’}$$ or xdx-ydy=0, and
$$\frac{dx}{zx}=\frac{dy}{xy’}$$ or ydx-zdy=0,
Integrating these we get two solutions
x2-y2=a , y2-z2=b
Hence, the general solution of the given equation is,
φ(x2-y2, y2-z2 )=0.

10. The equation 2$$\frac{dy}{dx} – xy = y^{-2},$$ is an example for Bernoulli’s equation.
a) False
b) True

Explanation: A first order, first degree differential equation of the form,
$$\frac{dy}{dx} + P(x). y = Q(x). y^a,$$ is known as Bernoulli’s equation.

11. A particular solution for an equation is derived by eliminating arbitrary constants.
a) True
b) False

Explanation: A particular solution for an equation is derived by substituting particular values to the arbitrary constants in the complete solution.

12. A partial differential equation is one in which a dependent variable (say ‘y’) depends on one or more independent variables (say ’x’, ’t’ etc.)
a) False
b) True

Explanation: A differential equation is divided into two types, ordinary and partial differential equations.
A partial differential equation is one in which a dependent variable depends on one or more independent variables.
Example: $$F(x,t,y,\frac{∂y}{∂x},\frac{∂y}{∂t},……)= 0$$

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