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Have you ever considered what it would take for doctors to measure infection rates and implement proper safety measures quickly if a deadly disease like COVID-19 were to reappear? Have you ever wondered how a hot cup of tea or coffee cools down in your room at a normal temperature? These are the situations in which knowledge of Differential equations becomes relevant, and in the NCERT Class 12 Maths, we will study this in detail. The main purpose of these NCERT Notes of the Integrals class 12 PDF is to provide students with an efficient study material from which they can revise the entire chapter.
Differential equations NCERT Class 12 Maths Notes cover topics like general and particular solutions of a differential equation, formation of differential equations, some methods to solve a first-order first-degree differential equation, and some applications of differential equations in different areas. These differential equations class 12 notes are prepared by Careers360 subject matter experts who have multiple years of experience in this field. For the syllabus, solutions, and chapter-wise PDFs, head over to this link: NCERT.
Use the link below to download the PDF version of the Integrals NCERT Notes for free. After that, you can view the PDF anytime you desire without internet access. It is very useful for revision and last-minute studies.
The basic definition of a differential equation is that it is a kind of equation that involves the differentiation of a function or functions.
For a given function g, find the function f such that
The above equation is called a differential equation.
An example of a general differential equation is:
This is an ordinary Differential equation.
General notation:
The first order differential equation is defined by the equation
Its order is 1.
Its order is 2.
It is defined as the highest order derivative it contains. In other words, it is the power to which the highest order derivative is raised.
Let us see an example.
It has a degree of 1 because the power of the highest order is 1
Example:
Find the order and degree of the given equation
Solution:
The order of the equation is 2, and the degree of the equation is 1.
Let us consider a curve
And let
Thus the function is
Here, the function consists of 2 arbitrary constants, a and b, and they are known as general solutions of a differential equation. If the function does not contain any arbitrary constant is called a particular solution.
Formation of a Differential equation whose general solution is given
We know that
Represents a circle with centre at
On differentiating it, we get
This is a differential equation that represents a family of circles.
Now, in the equation
a) If the family of curves F depends on one parameter, then it is represented by
F(x, y, a) =0
On differentiating it, we get
F(x, y, y’)=0
b) If the family of curves F depends on two parameters, then it is represented by
F(x, y, a, b)
On differentiating it, we get,
F(x, y, y’, y’’)=0
Here, we will find 3 ways of solving the differential equation.
Method of variable separation
Method of homogeneous differential equations
Method of Linear differential equations
The differential of the form
Where
It is said to be in a variable separable form.
Rewrite the equation as
This process is separating the variables.
Now, integrating both sides, we get
By this, we get the solution of the differential equation.
Let’s look at one illustration for a better understanding.
Solution of the differential equation
Rewrite the differential equation as
Integrating both sides, we get
Let us consider a function.
Now if we replace
Generalizing
Now, in the case of differential equation,
A differential equation of the form
We make substitution,
Differentiating (2) with respect to x, we get
Substituting (3) in (1), we get,
Using variable separation in (4)
Now integrating (5), we get,
Therefore (6) gives the solution
Let us see an example.
Substituting
Now, integrating, we get,
The linear differential equations are those in which the variable and its derivative occur only in the first degree.
An equation of the form
Where P(x) and Q(x) are functions of x only or constant is called a linear equation of the first order.
To solve the differential equation (i),
multiply both sides of Eq (i) by
Integrating both sides, we get,
Which is the required solution of the given differential equation.
The term
Thus, we remember the solution of the above equation as
Note :
Sometimes, a given differential equation becomes linear if we take x as the dependent variable and y as the independent variable.
Differential equations are used in a variety of disciplines, such as biology, economics, physics, chemistry, and engineering.
Growth and Decay Problem:
Let the amount of substance (or population) that is either growing or decaying be denoted by N(t). If we assume the time rate of change of this amount of substance,
Where k is the constant of proportionality. We are assuming that N(t) is a differentiable, hence continuous, function of time.
Given below are some previous year question answers of various examinations from the NCERT class 12 chapter 9, Differential Equations:
Question 1:
Solution:
Given
upon differentiation, we get
atter differentiation again we get
Hence, the correct answer is
Question 2: The solution of the differential equation
Solution:
Upon integration of both sides,
Hence, the correct answer is
Question 3: The solution of the differential equation
Solution:
Hence, the correct answer is
All the links of chapter-wise notes for NCERT class 12 maths are given below:
After completing the NCERT textbook solutions, students can practice these NCERT Exemplar questions. These w
The following links will lead students to the solutions of other subject exercises. These well-structured solutions will be very useful for them.
Students can use the following links to check the latest CBSE syllabus and some reference books.
Exam Date:22 July,2025 - 29 July,2025
Exam Date:22 July,2025 - 28 July,2025
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