Differential Equations

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Class 12 Maths Chapter 9
Differential Equations
Important Questions

In a typical mathematics curriculum, Chapter 12 often covers Differential Equations. The important questions from this chapter usually focus on key concepts and techniques used in solving differential equations. Here are cbse class 12 maths differential equations important questions you should be familiar with Understanding Types of Solving First-Order and First-Degree Equations.

Table of Contents

Introduction

Differential equations are a fundamental aspect of mathematics with extensive applications in various fields like physics, engineering, economics, and biology. At their core, differential equations are mathematical equations that relate a function with its derivatives. In this chapter cbse class 12 maths differential equations, you will delve into foundational elements encompassing topics such as ,Linear vs. Nonlinear Differential EquationsUnderstanding concepts is crucial as they form the basis for much of mathematical analysis and are widely used in various fields, including science, engineering, economics, and more.

What are Differential Equations?

Differential equations are a type of mathematical equation used to describe various phenomena that involve change and are fundamental in the fields of engineering, physics, economics.differential equations involve a function and its derivatives. The function usually represents a physical quantity, while the derivatives represent rates of change of that quantity. Linear vs. Nonlinear:

Linear differential equations are those in which the function and its derivatives appear linearly (i.e., not raised to any power nor multiplied by each other).
Nonlinear differential equations involve the function or its derivatives in a nonlinear manner, such as being raised to a power or multiplied together.

Class 12 Differential Equations Important Questions and Answers

Q1. What is the order of the differential equation y’’ + 5y’ + 6 = 0?

Options

(a) 0
(b) 1
(c) 2
(d) 3

Ans. (c) 2
Explanation:
Differential equation y’’ + 5y’ + 6 = 0.
The highest order derivative present in the differential equation is y’’. Hence, the order is 2.

Q2: The number of arbitrary constants in the particular solution of a differential equation of third order is:

Options

(a) 3
(b) 2
(c) 1
(d) 0

Ans. (d) 0
Explanation:
Particular solution of differential equation of any order, does not have any arbitrary constant.

Q3. Verify that the function y = a cos x + b sin x, where, a, b ∈ R is a solution of the differential equation

$\dfrac{d^2y}{dx^2} + y = 0.$

Explanation:
The given function is y = a cos x + b sin x … (1)
Differentiating both sides of equation (1) with respect to x,
$\dfrac{dy}{dx}$ = – a sin x + b cos x
$\\[4.5 bp] \dfrac{d^2y}{dx^2}$ = – a cos x – b sin x
LHS = $\dfrac{d^2y}{dx^2}$ + y
= – a cos x – b sin x + a cos x + b sin x = 0 = RHS
Hence, the given function is a solution to the given differential equation.

Q4. Find the differential equation of the family of lines through the origin.

Explanation:
y = mx be the family of lines through the origin.
Therefore, $\dfrac{dy}{dx} \text{= m} \\[2.5 bp]$ Eliminating m, (substituting m = $\dfrac{y}{x} )\\[4.5 bp]$ $\text{y =} \left(\dfrac{dy}{dx}\right) .\text{ x}\\[4.5 bp] \text{x. }\dfrac{dy}{dx}\text{ – y = 0}$

Q5: Find the integrating factor of

$x.\dfrac{dy}{dx}\text{ - 3y = x}^3$

Explanation:
Here $P = - \dfrac{3}{x}, \text{Q = x}^2 \\[4.5 bp]$ Integrating factor= $e^{∫\text{P dx}} \\[4.5 bp] = e^{∫-\frac{3}{x} dx}\\[4.5 bp] = e^{–\text{3 log x}}\\[4.5 bp] = x^{–3}$

CBSE Class 12 Maths Chapter wise Important Questions

Chapter No.	Chapter Name
Chapter 1	Relations and Functions
Chapter 2	Inverse Trigonometric Functions
Chapter 3	Matrices
Chapter 4	Determinants
Chapter 5	Continuity and Differentiability
Chapter 6	Applications of Derivatives
Chapter 7	Integrals
Chapter 8	Applications of the Integrals
Chapter 9	Differential Equations
Chapter 10	Vectors
Chapter 11	Three - dimensional Geometry
Chapter 12	Linear Programming
Chapter 13	Probability

Conclusion

Differential equations involves finding a solution that satisfies the equation and any given initial or boundary conditions.Exploring cbse class 12 maths differential equation important questions, becomes essential for a comprehensive grasp of this fundamental aspect of mathematics. To enhance understanding, platforms like Oswal.io provide a wealth of resources, including question-answer sets, comprehensive class 12 questions and answers.Utilising these resources ensures students acquire the knowledge necessary to excel in their studies and examinations, inviting them to delve deeper into the captivating realm of differential equations.

Frequently Asked Questions

Q1 : What is a differential equation?

Ans: A differential equation is a mathematical equation that relates a function with its derivatives. It is used to model the rate of change of a physical quantity, like velocity, temperature, or concentration over time or space.

Q2 : What are the types of differential equations?

Ans: There are mainly two types: Ordinary Differential Equations (ODEs), which involve functions of a single variable and their derivatives, and Partial Differential Equations (PDEs), which involve functions of multiple variables and their partial derivatives.

Q3 : How are differential equations solved?

Ans: Solutions can be found analytically using various techniques like separation of variables, integrating factors, or characteristic equations. When an analytical solution is not possible, numerical methods like Euler’s method, Runge-Kutta methods, or finite difference methods are used.

Q4 : What are initial and boundary conditions in differential equations?

Ans: Initial conditions specify the value of the function (and possibly its derivatives) at a specific point, usually at the start of the interval of interest. Boundary conditions specify the behavior of the solution at the boundaries of the domain, such as at the endpoints of an interval for ODEs or at the edges of a spatial domain for PDEs.

Q5: What is a linear differential equation?

Ans: A linear differential equation is one where the function and its derivatives appear linearly (i.e., raised to the first power and not multiplied or divided by each other). These equations are generally easier to solve than nonlinear differential equations.

Chapter Wise Important Questions for CBSE Board Class 12 Maths
Relations and Functions
Inverse Trigonometric Functions
Matrices
Determinants
Continuity and Differentiability
Applications of Derivatives
Integrals
Applications of the Integrals
Differential Equations
Vectors
Three - dimensional Geometry
Linear Programming
Probability

Class 12 Maths Chapter 9 Differential Equations Important Questions

Introduction

What are Differential Equations?

Class 12 Differential Equations Important Questions and Answers

Q1. What is the order of the differential equation y’’ + 5y’ + 6 = 0?

Options

(a) 0 (b) 1 (c) 2 (d) 3

Q2: The number of arbitrary constants in the particular solution of a differential equation of third order is:

Options

(a) 3 (b) 2 (c) 1 (d) 0

Q3. Verify that the function y = a cos x + b sin x, where, a, b ∈ R is a solution of the differential equation

\dfrac{d^2y}{dx^2} + y = 0.

Q4. Find the differential equation of the family of lines through the origin.

Q5: Find the integrating factor of

x.\dfrac{dy}{dx}\text{ - 3y = x}^3

CBSE Class 12 Maths Chapter wise Important Questions

Conclusion

Frequently Asked Questions

Q1 : What is a differential equation?

Q2 : What are the types of differential equations?

Q3 : How are differential equations solved?

Q4 : What are initial and boundary conditions in differential equations?

Q5: What is a linear differential equation?

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Class 12 Maths Chapter 9
Differential Equations
Important Questions

(a) 0
(b) 1
(c) 2
(d) 3

(a) 3
(b) 2
(c) 1
(d) 0

$\dfrac{d^2y}{dx^2} + y = 0.$

$x.\dfrac{dy}{dx}\text{ - 3y = x}^3$