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

1. Which of the following is an example of non-linear differential equation?

a) y=mx+c

b) x+x’=0

c) x+x^{2}=0

d) x”+2x=0

View Answer

Explanation: For a differential equation to be linear the dependent variable should be of first degree. Since in equation x+x

^{2}=0, x

^{2}is not a first power, it is not an example of linear differential equation.

2. Which of the following is not a standard method for finding the solutions for differential equations?

a) Variable Separable

b) Homogenous Equation

c) Orthogonal Method

d) Bernoulli’s Equation

View Answer

Explanation: The following are the different standard methods used in finding the solution of a differential equation:

- Variable Separable
- Homogenous Equation
- Non-homogenous Equation reducible to Homogenous Equation
- Exact Differential Equation
- Non-exact Differential Equation that can be made exact with the help of integrating factors
- Linear First Order Equation
- Bernoulli’s Equation

3. Solution of a differential equation is any function which satisfies the equation.

a) True

b) False

View Answer

Explanation: A solution of a differential equation is any function which satisfies the equation, i.e., reduces it to an identity. A solution is also known as integral or primitive.

4. A solution which does not contain any arbitrary constants is called a general solution.

a) True

b) False

View Answer

Explanation: The solution of a partial differential equation obtained by eliminating the arbitrary constants is called a general solution.

5. 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

View Answer

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

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

a) True

b) False

View Answer

Explanation: A solution which does not contain any arbitrary constants is called a general solution whereas a particular solution is derived by substituting particular values to the arbitrary constants in this solution.

7. Singular solution of a differential equation is one that cannot be obtained from the general solution gotten by the usual method of solving the differential equation.

a) True

b) False

View Answer

Explanation: A differential equation is said to have a singular solution if in all points in the domain of the equation the uniqueness of the solution is violated. Hence, this solution cannot be obtained from the general solution.

8. Which of the following equations represents Clairaut’s partial differential equation?

a) z=px+f(p,q)

b) z=f(p,q)

c) z=p+q+f(p,q)

d) z=px+qy+f(p,q)

View Answer

Explanation: Equations of the form, z=px+qy+f(p,q) are known as Clairaut’s partial differential equations, named after the Swiss mathematician, A. C. Clairaut (1713-1765).

9. Which of the following represents Lagrange’s linear equation?

a) P+Q=R

b) Pp+Qq=R

c) p+q=R

d) Pp+Qq=P+Q

View Answer

Explanation: Equations of the form, Pp+Qq=R are known as Lagrange’s linear equations, named after Franco-Italian mathematician, Joseph-Louis Lagrange (1736-1813).

10. A partial differential equation is one in which a dependent variable (say ‘x’) depends on an independent variable (say ’y’).

a) False

b) True

View Answer

Explanation: An ordinary 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: \(F(x,t,y,\frac{∂y}{∂x},\frac{∂y}{∂t},……)= 0. \)

11. What is the complete solution of the equation, \(q= e^\frac{-p}{α}\)?

a) \(z=ae^\frac{-a}{α}y\)

b) \(z=x+e^\frac{-a}{α}y\)

c) \(z=ax+e^\frac{-a}{α} y+c\)

d) \(z=e^\frac{-a}{α}y\)

View Answer

Explanation: Given: \(q= e^\frac{-p}{α}\)

The given equation does not contain x, y and z explicitly.

Setting p = a and q = b in the equation, we get \(b= e^\frac{-a}{α}.\)

Hence, a complete solution of the given equation is,

\(z=ax+by+c,\,with \, b= e^\frac{-a}{α}\)

\(z=ax+e^\frac{-a}{α} y+c.\)

12. A particular solution for an equation is derived by eliminating arbitrary constants.

a) True

b) False

View Answer

Explanation: A particular solution for an equation is derived by substituting particular values to the arbitrary constants in the complete solution thereby eliminating any arbitrary constants present in the solution. Such solution represents a particular member of the family of surfaces given by the complete solution.

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