# Ordinary Differential Equations Questions and Answers – Basic Definitions

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This set of Ordinary Differential Equations Multiple Choice Questions & Answers (MCQs) focuses on “”.

1. Which of the following correctly defines ordinary differential equations?
a) A differential equation in which a dependent variable (say ‘y’) depends on only one independent variable (say ’x’)
b) A differential equation in which an independent variable (say ‘y’) depends on only one dependent variable (say ’x’)
c) A differential equation in which a dependent variable (say ‘y’) depends on one or more independent variables (say ’x’, ’t’ etc.)
d) A differential equation in which an independent variable (say ‘y’) depends on one or more dependent variables (say ’x’, ’t’ etc.)

Explanation: A differential equation is an equation involving an unknown function y of one or more independent variables x, t, …… and its derivatives. These are divided into two types, ordinary or partial differential equations.
An ordinary differential equation is a differential equation in which a dependent variable (say ‘y’) is a function of only one independent variable (say ‘x’).

2. 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: 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: $$F(x,t,y,\frac{∂y}{∂x},\frac{∂y}{∂t},……)= 0$$

3. What is the order of the differential equation, y”+y’-x3y=sinx?
a) 2
b) 1
c) 0
d) 3

Explanation: Order of a differential equation is given by the highest order derivative appearing in the differential equation. Hence for the given equation, y”+y’-x3y=sinx, the order is 2.

4. What is the degree of the differential equation, 4x3-6x2 y3+2y=0?
a) 3
b) 5
c) 1
d) 8

Explanation: The degree of an equation that has not more than one variable in each term is the exponent of the highest power to which that variable is raised in the equation. But when more than one variable appears in a term, it is necessary to add the exponents of the variables within a term to get the degree of the equation. Hence, the degree of the equation, 4x3-6x2 y3+2y=0, is 2+3 = 5.

5. Which one of the following is not a criterion for linearity of an ordinary differential equation?
a) The dependent variable y and its derivatives are of first degree
b) The derivatives of the dependent variable y should be of second degree
c) No product terms of y and/or any of its derivatives are present
d) No transcendental functions of y and/or its derivatives occur

Explanation: The criterions for linearity of an ordinary differential equation are:

• The dependent variable y and its derivatives are of first degree
• No product terms of y and/or any of its derivatives are present
• No transcendental functions of y and/or its derivatives occur

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 not an example of linear differential equation?
a) y=mx+c
b) x+x’=0
c) x+x2=0
d) x”+2x=0

Explanation: For a differential equation to be linear the dependent variable should be of first degree. Since in equation x+x2=0, x2 is not a first power, it is not an example of linear differential equation.

8. Which of the following is not a standard method for finding the solutions for differential equations?
a) Variable Separable
b) Homogenous Equation
c) Bernoulli’s Equation
d) Orthogonal Method

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

9. Solution of a differential equation is any function which satisfies the equation.
a) False
b) True

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.

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

$$\frac{dy}{dx} + P(x). y = Q(x). y^a,$$ is known as Bernoulli’s equation. 