CBSE Class 12 Mathematics Notes Chapter 9 Differential Equations

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A differential equation is a type of equation that involves a relationship between one or more unknown functions and their derivatives.

Derivatives refer to the rate at which the function varies with respect to its independent variables.
Functions is a branch of mathematics that specifies relationships between physical quantities.
Differential equations are used to study different sets of solutions and the properties of their solutions.
It specifies the rate of change of the dependent variable with respect to the independent variable (variables).
Ordinary derivatives and partial derivatives are two types of derivatives used in the differential equation.
The concept is used in the fields of economics, engineering, physics and biology.
It can be represented as:

dy/dx = f(x)

According to the CBSE Syllabus 2023-24, the chapter on Differential Equations comes under Unit 3 of Calculus. Unit Calculus holds a weightage of around 35 marks in the NCERT Class 12 Mathematics examination.

Read More:

Differential Equation Preparation Resources
Differential Equation	NCERT Solutions For Class 12 Mathematics Chapter 9: Differential Equations
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Order of Differential Equation

The order of the differential equation refers to the highest order of the derivative of the equation.

It is a positive integer that depends on the derivative of an equation.
First order differential equations are linear equations that are expressed in the first order.
The second order differential equation includes second-order derivative equations.

Example of Order of Differential Equation Example: dy/dx = x + 9 , The order is 1.

Order of Differential Equations

Order of Differential Equations

Degree of Differential Equation

The degree of a differential equation refers to the power of the highest-order derivative when the equation is represented in polynomial form.
It is expressed in the form y’,y’’,y’’.

Example of Degree of Differential Equation Example: dy/dx + 7 = 0, Its degree is 1.

Degree of Differential Equations

Degree of Differential Equations

General and Particular Solutions of a Differential Equation

The general solutions of a differential equation include arbitrary constants (in the case of an ODE) or arbitrary functions (in the case of a PDE.)
ODE stands for ordinary differential equations, and PDE stands for partial differential equations.
In a general solution, the number of arbitrary constants is the same as the order of a given differential equation.
Particular solutions of a differential equation are obtained from a general solution when arbitrary constants are converted into required values.
Depending on the values of arbitrary constants, there are numerous particular solutions of a differential equation.

General and Particular Solutions of a Differential Equation

General and Particular Solutions of a Differential Equation

Solution of Differential Equations by Method of Separation of Variables

The solution of a differential equation, which is also known as primitivity, expresses the relationship between the variables and independent of derivatives.
In this process, the x-terms and y-terms of differential equations are separated into different sides, which also include the delta terms.
The separated variable is then integrated to derive the solution of the differential equation.
It is expressed in the form:

f(x)d(x) + g(y)d(y) = 0

Separation of Variables

Separation of Variables

Solutions of Homogeneous Differential Equations

Homogeneous differential equations are a type of differential equation where the degree of f(x,y) and g(x, y) is the same.
It can be written in the form kn F(x,y) where k≠0.
It has multiplicative scaling behaviour, which means when all arguments are multiplied by a factor, the function value is also multiplied by the power of the factor.

Steps to solve Homogeneous Differential Equations

To solve these equations, first substitute the value of y in the given differential equation.
Next, differentiate the equation followed by the separation of variables.
Lastly, integrate both sides of the equation.

Homogeneous Equation

Homogeneous Equation

Linear Differential Equations

A linear differential equation represents a differential equation which consists of a variable, a derivative of this variable, and a few other functions.
It is also known as Linear Partial Differential Equation in case the function is dependent on variables and derivatives are partial.
These equations form linear polynomial equations.
The equation of the form dy/dx + Py = Q is called a first-order linear differential equation where P and Q are constants.

How to Solve First Order Linear Differential Equation

First, organize the terms in the form of dy/dx + Py = Q.
Obtain the integrating factor by integrating P with respect to x and raise the power of x.
Multiplying both sides of the differential equation with an integrating factor.
The left-hand side of the equation is the derivative of y × M (x).
Lastly, integrate both sides of the equation with respect to x and obtain the constant value.

Linear Differential Equations

Linear Differential Equations

There are Some important List Of Top Mathematics Questions On Differential Equations Asked In CBSE CLASS XII

CBSE CLASS XII Related Questions

1.
If $ \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 $, $ |\overrightarrow{a}| = \sqrt{37} $, $ |\overrightarrow{b}| = 3 $, and $ |\overrightarrow{c}| = 4 $, then the angle between $ \overrightarrow{b} $ and $ \overrightarrow{c} $ is:

$ \frac{\pi}{6} $
$ \frac{\pi}{4} $
$ \frac{\pi}{3} $
$ \frac{\pi}{2} $

View Solution

2.
Draw a rough sketch for the curve $y = 2 + |x + 1|$. Using integration, find the area of the region bounded by the curve $y = 2 + |x + 1|$, $x = -4$, $x = 3$, and $y = 0$.

View Solution

3.
The integrating factor of the differential equation $ (x + 2y^3) \frac{dy}{dx} = 2y $ is:

$ e^{y^2} $
$ \frac{1}{\sqrt{y}} $
$ e^{-\frac{1}{y^2}} $
$ e^{y^2} $

View Solution

4.
Evaluate : \[ I = \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}} \]

View Solution

5.
Let $|\vec{a}| = 5 \text{ and } -2 \leq \lambda \leq 1$. Then, the range of $|\lambda \vec{a}|$ is:

[5, 10]
[-2, 5]
[-1, 5]
[10, 5]

View Solution

6.
Let both $AB'$ and $B'A$ be defined for matrices $A$ and $B$. If the order of $A$ is $n \times m$, then the order of $B$ is:

$n \times n$
$n \times m$
$m \times m$
$m \times n$

View Solution

CBSE Class 12 Mathematics Notes Chapter 9 Differential Equations

Order of Differential Equation

Example of Order of Differential Equation

Degree of Differential Equation

Example of Degree of Differential Equation

General and Particular Solutions of a Differential Equation

Solution of Differential Equations by Method of Separation of Variables

Solutions of Homogeneous Differential Equations

Linear Differential Equations

How to Solve First Order Linear Differential Equation

CBSE CLASS XII Related Questions

1.
If \( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 \), \( |\overrightarrow{a}| = \sqrt{37} \), \( |\overrightarrow{b}| = 3 \), and \( |\overrightarrow{c}| = 4 \), then the angle between \( \overrightarrow{b} \) and \( \overrightarrow{c} \) is:

2.
Draw a rough sketch for the curve $y = 2 + |x + 1|$. Using integration, find the area of the region bounded by the curve $y = 2 + |x + 1|$, $x = -4$, $x = 3$, and $y = 0$.

3.
The integrating factor of the differential equation \( (x + 2y^3) \frac{dy}{dx} = 2y \) is:

4.
Evaluate : \[ I = \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}} \]

5.
Let $|\vec{a}| = 5 \text{ and } -2 \leq \lambda \leq 1$. Then, the range of $|\lambda \vec{a}|$ is:

6.
Let both $AB'$ and $B'A$ be defined for matrices $A$ and $B$. If the order of $A$ is $n \times m$, then the order of $B$ is:

Similar Mathematics Concepts

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CBSE Class 12 Mathematics Notes Chapter 9 Differential Equations

Order of Differential Equation

Example of Order of Differential Equation

Degree of Differential Equation

Example of Degree of Differential Equation

General and Particular Solutions of a Differential Equation

Solution of Differential Equations by Method of Separation of Variables

Solutions of Homogeneous Differential Equations

Linear Differential Equations

How to Solve First Order Linear Differential Equation

CBSE CLASS XII Related Questions

1.If \( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 \), \( |\overrightarrow{a}| = \sqrt{37} \), \( |\overrightarrow{b}| = 3 \), and \( |\overrightarrow{c}| = 4 \), then the angle between \( \overrightarrow{b} \) and \( \overrightarrow{c} \) is:

2.Draw a rough sketch for the curve $y = 2 + |x + 1|$. Using integration, find the area of the region bounded by the curve $y = 2 + |x + 1|$, $x = -4$, $x = 3$, and $y = 0$.

3.The integrating factor of the differential equation \( (x + 2y^3) \frac{dy}{dx} = 2y \) is:

4. Evaluate : \[ I = \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}} \]

5.Let $|\vec{a}| = 5 \text{ and } -2 \leq \lambda \leq 1$. Then, the range of $|\lambda \vec{a}|$ is:

6.Let both $AB'$ and $B'A$ be defined for matrices $A$ and $B$. If the order of $A$ is $n \times m$, then the order of $B$ is:

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
If \( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 \), \( |\overrightarrow{a}| = \sqrt{37} \), \( |\overrightarrow{b}| = 3 \), and \( |\overrightarrow{c}| = 4 \), then the angle between \( \overrightarrow{b} \) and \( \overrightarrow{c} \) is:

2.
Draw a rough sketch for the curve $y = 2 + |x + 1|$. Using integration, find the area of the region bounded by the curve $y = 2 + |x + 1|$, $x = -4$, $x = 3$, and $y = 0$.

3.
The integrating factor of the differential equation \( (x + 2y^3) \frac{dy}{dx} = 2y \) is:

4.
Evaluate : \[ I = \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}} \]

5.
Let $|\vec{a}| = 5 \text{ and } -2 \leq \lambda \leq 1$. Then, the range of $|\lambda \vec{a}|$ is:

6.
Let both $AB'$ and $B'A$ be defined for matrices $A$ and $B$. If the order of $A$ is $n \times m$, then the order of $B$ is: