Class 12 Chapter 9

Class 12 Maths Chapter 9: Differential Equations

A Differential Equation is an equation involving a dependent variable, an independent variable, and one or more of its derivatives. It provides a mathematical model for describing changing quantities and is widely used in Physics, Engineering, Biology, and Economics. This chapter introduces the formation, order, degree, and solution of first-order differential equations using standard methods. It is an important topic in CBSE Board, JEE Main, and JEE Advanced, combining conceptual understanding with analytical problem-solving.

At ALLEN, Class 12 Maths Chapter 9: Differential Equations smart study resuorces simplify Differential Equations through concise theory, important formulas, standard solving techniques, solved examples, and exam-focused strategies. Class 12 Maths Chapter 9 revision notes are ideal for quick revision and building confidence for both board and competitive examinations.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate
Revision Time	3–4 Hours
Key Concepts Covered	Order and Degree of Differential Equation, General and Particular Solution, Homogeneous Differential Equations

2.0Chapter 9 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution
Practice Test Paper with Answers

Test your Knowledge

question 1 of 5

The order of a differential equation is the:

1.Number of terms in the equation

2.Number of variables

3.Highest order derivative present

4.Degree of the equation

Must-Revise Concepts

Differential Equation PYQ with solutions

Differential Equations

Exact Differential Equation

First order Differential Equation

Homogeneous Differential Equation

Homogeneous differential equations

Linear differential equations

Class 12 Maths Chapters List:- JEE & NEET Revision Notes

Chapter 1 Relations and Functions

Chapter 2 Inverse Trigonometric Functions

Chapter 3 Matrices

Chapter 4 Determinants

Chapter 5 Continuity and Differentiability

Chapter 6 Application of Derivatives

Chapter 7 Integrals

Chapter 8 Application of Integrals

Chapter 9 Differential Equations

Chapter 10 Vector Algebra

Chapter 11 Three Dimensional Geometry

Chapter 12 Linear Programming

Chapter 13 Probability

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JEE Maths

Class 12 Chapter 9

Class 12 Maths Chapter 9: Differential Equations

A Differential Equation is an equation involving a dependent variable, an independent variable, and one or more of its derivatives. It provides a mathematical model for describing changing quantities and is widely used in Physics, Engineering, Biology, and Economics. This chapter introduces the formation, order, degree, and solution of first-order differential equations using standard methods. It is an important topic in CBSE Board, JEE Main, and JEE Advanced, combining conceptual understanding with analytical problem-solving.

At ALLEN, Class 12 Maths Chapter 9: Differential Equations smart study resuorces simplify Differential Equations through concise theory, important formulas, standard solving techniques, solved examples, and exam-focused strategies. Class 12 Maths Chapter 9 revision notes are ideal for quick revision and building confidence for both board and competitive examinations.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate
Revision Time	3–4 Hours
Key Concepts Covered	Order and Degree of Differential Equation, General and Particular Solution, Homogeneous Differential Equations

2.0Chapter 9 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution
Practice Test Paper with Answers

Must-Revise Concepts

Differential Equation PYQ with solutions

Differential Equations

Exact Differential Equation

First order Differential Equation

Homogeneous Differential Equation

Homogeneous differential equations

Linear differential equations

Differential Equations

An equation that involves independent and dependent variables and the derivatives of the dependent variables is called a Differential Equation (D. E).

A Differential Equation (D. E) is said to be ordinary, if the differential coefficients have reference to a single independent variable only e.g. $\frac{d ^{2} y}{d x ^{2}} - \frac{2 d y}{d x} + cos x = 0$ and it is said to be partial if there are two or more independent variables. e.g. $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 0$ is a partial differential equation. We are concerned with ordinary differential equations only.

1.0Order Of Differential Equations

The order of a differential equation (D.E) is the order of the highest differential coefficient occurring in it.

Example:

$\frac{d ^{2} y}{dx ^{2}} + \frac{dy}{dx} + 1 = 0$ , Order is 2

$\frac{d ^{3} y}{d x ^{3}} + x = 0$ , Order is 3.

2.0Degree of Differential Equations

The highest exponent of the highest order differential coefficient, when the differential equation is expressed as a polynomial in all the differential coefficients, is called degree of the differential equation.

Thus, the differential equation:

$a_{0, 0} + \sum_{i = 1}^{n_{1}} a_{i, 1} (\frac{d ^{m} y}{d x ^{m}})^{i} + \sum_{i = 1}^{n_{2}} a_{i, 2} (\frac{d ^{m - 1} y}{d x ^{m - 1}})^{i} + \dots = 0$ is of order m & degree $n_{1} .$

where $a_{i, j}$ is some expression in x, y and $a_{n_{1}, 1} \neq = 0$

Example: Find degree of $\frac{d ^{2} y}{d x ^{2}} = 3 \frac{d y}{d x} + 4$

Solution: $(\frac{d ^{2} y}{d x ^{2}})^{3} = \frac{d y}{d x} + 4$

Degree = 3

1.0Solution Of Differential Equations

The solution to a differential equation is a relation between the variables that does not include derivatives but satisfies the given differential equation. In other words, it is a function from which the given differential equation can be derived.

Thus, the solution of $\frac{d y}{d x} = e^{x}$ could be obtained by simply integrating both sides, i.e., $y = e^{x} + c$ and that of, $\frac{d y}{d x} = p x + q i sy = \frac{p x ^{2}}{2} + q x + c$ , where c is an arbitrary constant.

(i) A general solution, or integral, of a differential equation is a relation between the variables that does not include derivatives. This relation contains as many arbitrary constants as the order of the differential equation.

For example, a general solution of the differential equation $\frac{d ^{2} x}{d t ^{2}} = - 4 x i s x = A cos 2 t + B sin 2 t$

where A and B are the arbitrary constants.

(ii) Particular solution or integral is that solution of the differential equation which is obtained from the general solution by assigning particular values to the arbitrary constant in the general solution.

For example, $x = 10 cos 2 t + 5 s in 2 t$ is a particular solution of differential equation $\frac{d ^{2} x}{d t ^{2}} = - 4 x$ .

2.0Elementary Types of First Order and First-Degree Differential Equations

Separation of Variables

Some differential equations can be solved by the method of separation of variables (or “variable separable”).

This method is only possible, if we can express the differential equation in the form

$A (x) d x + B (y) d y = 0$

where A(x) is a function of '' only and B(y) is a function of $’ y’$ only.

A general solution of this is given by,

$\int A (x) d x + \int B (y) d y = c$

where $’C’$ is the arbitrary constant.

Homogeneous Differential Equations

A function f(x, y) is said to be a homogeneous function of degree n, if the substitution

$x = λ x, y = λ y, λ > 0$ produces the equality

$f_{(λ^{x}, λ^{y})} = λ^{n} f (x, y)$

The degree of homogeneity $’n’$ can be any real number.

A differential equation of the form $\frac{d y}{d x} = \frac{f ( x , y )}{g ( x , y )}$

Where f(x, y) and g(x, y) are homogeneous functions of x, y and of the same degree are said to be homogeneous. Such equations can be solved by substituting y = vx.

Equation Reducible to Homogeneous form

The equation of the form $\frac{d y}{d x} = \frac{a _{1} x + b _{1} y + c _{1}}{a _{2} x + b _{2} y + c _{2}} Wh ere \frac{a _{1}}{a _{2}} \neq = \frac{b _{1}}{b _{2}}$ can be reduced to homogeneous form by changing the variable x, y to u, v as x = u + h, y = v + k.

Also Read Homogeneous Differential Equation

Linear Differential Equations

A differential equation is considered linear if the dependent variable and its derivatives appear only to the first power and are not multiplied by each other.

The n^th order linear differential equation is of the form:

$a_{0} (x) \frac{d ^{n} y}{d x ^{n}} + a_{1} (x) \frac{d ^{n - 1} y}{d x ^{n - 1}} + \dots\dots .. + a_{n} (x) y = φ (x), w h ere a_{0} (x), a_{1} (x) \dots . a_{n} (x)$ are called the coefficients of the differential equation.

Note that a linear differential equation is always of the first degree. However, not every differential equation of the first degree. e.g. the differential equation $\frac{d ^{2} y}{d x ^{2}} + (\frac{d y}{d x})^{3} + y^{2} = 0$ is not linear, though its degree is 1.

(i) Linear Differential Equations of First Order:

The most general form of a linear differential equation of first order is $\frac{d y}{d x} + P y = Q$ , where $P_{&} Q$

are functions of x.

To solve such an equation multiply both sides by $e^{\int P d x}$ .

So that we get $e^{\int P d x} [\frac{d y}{d x} + P y] = Q e^{\int P d x}$ ...(i)

$\frac{d}{d x} (e^{\int P d x} \cdot y) = Q e^{\int P d x}$ ...(ii)

On integrating equation (ii), we get

$y e^{\int P d x} = \int Q e^{\int P d x} d x + c$

This is the required general solution.

Also Read: First Order Differential Equation

(ii) Equation Reducible to Linear Form:

The equation of the form $\frac{d y}{d x} + P y = Q y^{n}$ , where P and Q are functions of x, is called Bernoulli’s equation.

On dividing by $y^{n}$ , we get $y^{- n} \frac{d y}{d x} + P y^{- n + 1} = Q$

Let $y^{- n + 1} = t, so t ha t (- n + 1) y^{- n} \frac{d y}{d x} = \frac{d t}{d x}$

then equation becomes $\frac{d t}{d x} + P (1 - n) t = Q (1 - n)$

which is linear with t as a dependent variable.

3.0Solved Examples of Differential Equations

Example 1: Find the order and degree of the following differential equation: $\frac{d ^{2} y}{d x ^{2}} = sin (\frac{d y}{d x})$

Solution: The given differential equation has order 2. Since the given differential equation is not a polynomial in the differential coefficients, the degree of the equation is not defined.

Example 2: Solve the differential equation $x y \frac{d y}{d x} = \frac{1 + y ^{2}}{1 + x ^{2}} (1 + x + x^{2})$

Solution: Differential equation can be rewritten as

$x y = (1 + y^{2}) (1 + \frac{x}{1 + x ^{2}}) \Rightarrow \frac{y}{1 + y ^{2}} d y = (\frac{1}{x} + \frac{1}{1 + x ^{2}}) d x$

Integrating, we get

$\frac{1}{2} ln (1 + y^{2}) = ln x + tan^{- 1} x + ln c \Rightarrow 1 + y^{2} = c x e^{t a n^{- 1} x}$

Example 3: Solve $\frac{d y}{d x} = \frac{x - 2 y + 1}{2 x + 2 y + 3}$

Solution:

$\frac{d y}{d x} = \frac{x - 2 y + 1}{2 x + 2 y + 3}$

$⊨^{2 x d y + 2 y d y + 3 d y} = x d x - 2 y d x + d x$

$\Rightarrow (2 y + 3) d y = (x + 1) d x - 2 (x d y + y d x)$

On integrating, we get

$\Rightarrow \int (2 y + 3) d y = \int (x + 1) d x - \int 2 d (x y)$

Solving : $y^{2} + 3 y = \frac{x ^{2}}{2} + x - 2 x y + c$

Example 4: Find the degree of homogeneity of function

(i) $f (x, y) = x^{2} + y^{2}$

(ii) $f (x, y) = (x^{\frac{3}{2}} + y^{\frac{3}{2}}) / (x + y)$

(iii) $f (x, y) = sin (\frac{x}{y})$

Solution:

(i) $f (λ x, λ y) = λ^{2} x^{2} + λ^{2} y^{2} = λ^{2} (x^{2} + y^{2}) = λ^{2} f (x, y)$

degree of homogeneity ⟶ 2

(ii) $f (λ x, λ y) = \frac{λ ^{3/2} x ^{3/2} + λ ^{3/2} y ^{3/2}}{λ x + λ y} f (λ x, λ y) = λ^{\frac{1}{2}} f (x, y)$

degree of homogeneity ⟶ 1/2

(iii) $f (λ x, λ y) = sin (\frac{λ x}{λ y}) = λ^{\circ} sin (\frac{x}{y}) = λ^{\circ} f (x, y)$

degree of homogeneity ⟶ 0

Example 5: The solution of the given differential equation $\frac{d y}{d x} = \frac{s i n y + x}{s i n 2 y - x c o s y} s$ -

(A) $sin^{2} y = x sin y + \frac{x ^{2}}{2} + c$ (B) $sin^{2} y = x sin y - x + c$

(C) $sin^{2} y = x + sin y + x^{2} + c$ (D) $sin^{2} y = x - sin y + \frac{x ^{2}}{2} + c$

Ans. (A)

Solution:

Here, $\frac{d y}{d x} = \frac{s i n y + x}{s i n 2 y - x c o s y}$

$\Rightarrow (cos y) \frac{d y}{d x} = \frac{s i n y + x}{2 s i n y - x} (p u t sin y = t)$

$\Rightarrow \frac{d t}{d x} = \frac{t + x}{2 t - x}$ (put t=v x)

$\frac{x d v}{d x} + v = \frac{v + 1}{2 v - 1}$

$∴ \frac{x d v}{d x} = \frac{v + 1}{2 v - 1} - v = \frac{v + 1 - 2 v ^{2} + v}{2 v - 1}$

or $\frac{2 v - 1}{- 2 v ^{2} + 2 v + 1} d v = \frac{d x}{x}$

on solving, we get

$sin^{2} y = x sin y + + c$ .

Example 6: Solve $\frac{d y}{d x} = \frac{x + 2 y + 3}{2 x + 3 y + 4}$

Solution:

Equation reducible to homogeneous form

Put $x = X + h, y = Y + k 0$

We have $\frac{d Y}{d X} = \frac{X + 2 Y + ( h + 2 k + 3 )}{2 X + 3 Y + ( 2 h + 3 k + 4 )}$

To determine h and k, we write

$h + 2 k + 3 = 0, 2 h + 3 k + 4 = 0 \Rightarrow h = 1, k = - 2$

So that $\frac{d Y}{d X} = \frac{X + 2 Y}{2 X + 3 Y}$

Putting $Y = V X$ , we get

$V + X \frac{d V}{d X} = \frac{1 + 2 V}{2 + 3 V} \Rightarrow \frac{2 + 3 V}{3 V ^{2} - 1} d V = - (\frac{d X}{X})$

$\Rightarrow [\frac{2 + 3}{2 ( 3 V - 1 )} - \frac{2 - 3}{2 ( 3 V + 1 )}] d V = - (\frac{d X}{X})$

$\Rightarrow \frac{2 + 3}{2 3} lo g (3 V - 1) - \frac{2 - 3}{2 3} lo g (3 V + 1) = (- lo g X + c)$

$∣^{\frac{2 + 3}{2 3}} lo g (3 Y - X) - \frac{2 - 3}{2 3} lo g (3 Y + X) = A$

where A is another

constant and

$X = x - 1, Y = y + 2 .$

Example 7: Which of the following equations is a linear differential equation?

(A) $\frac{d y}{d x} + x y^{2} = 1$ (B) $x^{2} + y = e^{x}$

(C) $\frac{d y}{d x} + 3 y = x y^{2}$ (D) $x \frac{d y}{d x} + y^{2} = sin x$

Ans. (B)

Solution:

Clearly answer is (B)

Example 8: Solve the differential equation $\frac{d y}{d x} + \frac{y}{x} = x^{2} y^{6}$ .

Solution:

The provided differential equation can be written as $\frac{1}{y ^{6}} \frac{d y}{d x} + \frac{1}{x y ^{5}} = x^{2}$

Putting y^{-5}=v so that

$- 5 y^{- 6} \frac{d y}{d x} = \frac{d v}{d x} or y^{- 6} \frac{d y}{d x} = - \frac{1}{5} \frac{d v}{d x}$

we get

$- \frac{1}{5} \frac{d v}{d x} + \frac{1}{x} v = x^{2} \Rightarrow \frac{d v}{d x} - \frac{5}{x} v = - 5 x^{2}$ ...(i)

This is the standard form of the linear differential equation having integrating factor

I.F = $e^{\int - \frac{5}{x} d x} = e^{- 5 l o g x} = \frac{1}{x ^{5}}$

Multiplying both sides of (i) by I.F. and integrating w.r.t. x

We get $v. \frac{1}{x ^{5}} = \int - 5 x^{2} \cdot \frac{1}{x ^{5}} d x$

$\Rightarrow \frac{v}{x ^{5}} = \frac{5}{2} x^{- 2} + c$

$\Rightarrow y^{- 5} x^{- 5} = \frac{5}{2} x^{- 2} + c$

which is the required solution.

Also Read Differential Equation previous year questions with solutions

4.0Practice Question on Differential Equations

1. Find the order and degree of differential equation $[\frac{d ^{2} x}{d t ^{2}}]^{3} + [\frac{d x}{d t}]^{4} - x t = 0$

(A) order : 3 ; degree : 2 (B) order : 2 ; degree : 4

(C) order : 2 ; degree : 3 (D) order : 4 ; degree : 2

2. Obtain the differential equation of the family of circles $x^{2} + y^{2} + 2 gx + 2 f y + c = 0$ ; where $g, f & c$

are arbitrary constants.

(A) $[1 + (y^{'})^{2}] \cdot y^{'''} - 3 y^{'} (y^{''})^{2} = 0$

(B) $[1 + (y^{'})^{2}] \cdot y^{'''} + (y^{''} + 2 y^{'}) (y^{''})^{2} = 0$

(C) $[1 + (y^{'})^{2}] \cdot y^{'''} + 3 y^{'} (y^{''})^{2} = 0$

(D) $[1 - (y^{'})^{2}] \cdot y^{'''} + 3 y^{'} (y^{''})^{2} = 0$

3. Solution of the differential equation $\frac{d y}{d x} = sin (x + y) + cos (x + y)$ is

(A) $tan (\frac{x + y}{2}) = a e^{x} - 1$

(B) $tan (\frac{x - y}{2}) = a e^{x} - 1$

(C) $tan (\frac{x + y}{2}) = a e^{- x} - 1$

(D) $tan (\frac{x + y}{2}) = a e^{x} + 1$

4. Solution of the differential equation $\frac{x d x + y d y}{x ^{2} + y ^{2}} = \frac{x d y - y d x}{x ^{2}}$ is

(A) $x^{2} - y^{2} = \frac{y}{x} + c$

(B) $x^{2} + y^{2} = - \frac{y}{x} + c$

(C) $x^{2} + y^{2} = \frac{x}{y} + c$

(D) $x^{2} + y^{2} = \frac{y}{x} + c$

Answer Sheet:

Question	1	2	3	4
Answer	C	C	A	D

Also Read

Divergence and Curl	Differentiability	Applications of Derivatives
Divergence Theorem	Differential Calculus	Partial Differential Equations

1.0Learning Outcomes

After completing this chapter, you will be able to:

Understand the concept of differential equations.
Identify the order and degree of a differential equation.
Form differential equations from given functions.
Solve first-order first-degree differential equations.
Apply the method of variable separation.
Verify the solution of differential equations.
Solve JEE Main, JEE Advanced, and CBSE Board questions effectively.

2.05-Minute Quick Revision

Must-Revise Concepts

Differential Equation
Order of Differential Equation
Degree of Differential Equation
Formation of Differential Equations
General Solution
Particular Solution
Variable Separable Method
Homogeneous Differential Equations
Verification of Solutions
Applications of Differential Equations

Important Formula Recall

General form: $F (x, y, \frac{d y}{d x}) = 0$
Variable separable form: $\frac{d y}{d x} = f (x) g (y)$
Separated form: $\frac{d y}{g ( y )} = f (x), d x$

Important Properties / Shortcut Tips

Determine the order before solving the equation.
Simplify the equation before identifying its degree.
Check whether variables can be separated.
Always include the constant of integration.
Verify the obtained solution by differentiation whenever required.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Order & Degree	High	Easy
Formation of Differential Equations	High	Moderate
Variable Separable Method	Very High	Moderate
General & Particular Solutions	High	Moderate
Homogeneous Differential Equations	Very High	Moderate–High
Verification of Solutions	Moderate	Easy–Moderate
Mixed Differential Equation Problems	Very High	High

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

Common Mistake	Correct Approach
Confusing order with degree	Order depends on the highest derivative, while degree depends on its power after simplification.
Forgetting to separate variables correctly	Rearrange the equation completely before integrating.
Omitting the constant of integration	Always include + C in the general solution.
Incorrect integration	Apply the appropriate integration formula carefully.
Ignoring initial conditions	Use the given conditions to determine the particular solution.
Not verifying the final answer	Differentiate the solution and substitute it back into the original equation.
Algebraic simplification errors	Simplify expressions before and after integration.

6.0ALLEN Faculty Tips

Strengthen your integration concepts before studying this chapter.
Memorize the definitions of order and degree thoroughly.
Practice variable separable equations regularly.
Solve formation-based questions to improve conceptual understanding.
Verify your answers whenever possible.
Practice Previous Years' Questions (PYQs) for better speed and accuracy.

7.0PYQ Trend Analysis

Differential Equations is a frequently tested chapter in CBSE Board, JEE Main, and JEE Advanced. Questions generally focus on identifying order and degree, solving first-order differential equations, and applying the method of separation of variables.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Order & Degree	1	Easy
Formation of Differential Equations	1	Moderate
Variable Separable Equations	2–3	Moderate
General & Particular Solutions	1	Moderate
Mixed Differential Equation Problems	1	Moderate–High

8.0Most Common Question Types

Finding the order and degree of a differential equation.
Forming differential equations from given functions.
Solving variable separable differential equations.
Finding general and particular solutions.
Verifying whether a given function satisfies a differential equation.
Applying integration techniques to solve differential equations.
Solving concept-based and previous years' examination questions.

9.0Smart Revision Strategy

Revise integration formulas before starting this chapter.
Learn the concepts of order and degree thoroughly.
Practice formation of differential equations.
Solve variable separable equations.
Revise general and particular solutions.
Verify solved equations to improve accuracy.
Practice Previous Years' Questions (PYQs).
Complete revision with a timed chapter test.

Class 12 Maths Chapters List:- JEE & NEET Revision Notes

Chapter 1 Relations and Functions

Chapter 2 Inverse Trigonometric Functions

Chapter 3 Matrices

Chapter 4 Determinants

Chapter 5 Continuity and Differentiability

Chapter 6 Application of Derivatives

Chapter 7 Integrals

Chapter 8 Application of Integrals

Chapter 9 Differential Equations

Chapter 10 Vector Algebra

Chapter 11 Three Dimensional Geometry

Chapter 12 Linear Programming

Chapter 13 Probability

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Differential Equations

An equation that involves independent and dependent variables and the derivatives of the dependent variables is called a Differential Equation (D. E).

A Differential Equation (D. E) is said to be ordinary, if the differential coefficients have reference to a single independent variable only e.g. $\frac{d ^{2} y}{d x ^{2}} - \frac{2 d y}{d x} + cos x = 0$ and it is said to be partial if there are two or more independent variables. e.g. $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 0$ is a partial differential equation. We are concerned with ordinary differential equations only.

1.0Order Of Differential Equations

The order of a differential equation (D.E) is the order of the highest differential coefficient occurring in it.

Example:

$\frac{d ^{2} y}{dx ^{2}} + \frac{dy}{dx} + 1 = 0$ , Order is 2

$\frac{d ^{3} y}{d x ^{3}} + x = 0$ , Order is 3.

2.0Degree of Differential Equations

The highest exponent of the highest order differential coefficient, when the differential equation is expressed as a polynomial in all the differential coefficients, is called degree of the differential equation.

Thus, the differential equation:

$a_{0, 0} + \sum_{i = 1}^{n_{1}} a_{i, 1} (\frac{d ^{m} y}{d x ^{m}})^{i} + \sum_{i = 1}^{n_{2}} a_{i, 2} (\frac{d ^{m - 1} y}{d x ^{m - 1}})^{i} + \dots = 0$ is of order m & degree $n_{1} .$

where $a_{i, j}$ is some expression in x, y and $a_{n_{1}, 1} \neq = 0$

Example: Find degree of $\frac{d ^{2} y}{d x ^{2}} = 3 \frac{d y}{d x} + 4$

Solution: $(\frac{d ^{2} y}{d x ^{2}})^{3} = \frac{d y}{d x} + 4$

Degree = 3

1.0Solution Of Differential Equations

The solution to a differential equation is a relation between the variables that does not include derivatives but satisfies the given differential equation. In other words, it is a function from which the given differential equation can be derived.

Thus, the solution of $\frac{d y}{d x} = e^{x}$ could be obtained by simply integrating both sides, i.e., $y = e^{x} + c$ and that of, $\frac{d y}{d x} = p x + q i sy = \frac{p x ^{2}}{2} + q x + c$ , where c is an arbitrary constant.

(i) A general solution, or integral, of a differential equation is a relation between the variables that does not include derivatives. This relation contains as many arbitrary constants as the order of the differential equation.

For example, a general solution of the differential equation $\frac{d ^{2} x}{d t ^{2}} = - 4 x i s x = A cos 2 t + B sin 2 t$

where A and B are the arbitrary constants.

(ii) Particular solution or integral is that solution of the differential equation which is obtained from the general solution by assigning particular values to the arbitrary constant in the general solution.

For example, $x = 10 cos 2 t + 5 s in 2 t$ is a particular solution of differential equation $\frac{d ^{2} x}{d t ^{2}} = - 4 x$ .

2.0Elementary Types of First Order and First-Degree Differential Equations

Separation of Variables

Some differential equations can be solved by the method of separation of variables (or “variable separable”).

This method is only possible, if we can express the differential equation in the form

$A (x) d x + B (y) d y = 0$

where A(x) is a function of '' only and B(y) is a function of $’ y’$ only.

A general solution of this is given by,

$\int A (x) d x + \int B (y) d y = c$

where $’C’$ is the arbitrary constant.

Homogeneous Differential Equations

A function f(x, y) is said to be a homogeneous function of degree n, if the substitution

$x = λ x, y = λ y, λ > 0$ produces the equality

$f_{(λ^{x}, λ^{y})} = λ^{n} f (x, y)$

The degree of homogeneity $’n’$ can be any real number.

A differential equation of the form $\frac{d y}{d x} = \frac{f ( x , y )}{g ( x , y )}$

Where f(x, y) and g(x, y) are homogeneous functions of x, y and of the same degree are said to be homogeneous. Such equations can be solved by substituting y = vx.

Equation Reducible to Homogeneous form

The equation of the form $\frac{d y}{d x} = \frac{a _{1} x + b _{1} y + c _{1}}{a _{2} x + b _{2} y + c _{2}} Wh ere \frac{a _{1}}{a _{2}} \neq = \frac{b _{1}}{b _{2}}$ can be reduced to homogeneous form by changing the variable x, y to u, v as x = u + h, y = v + k.

Also Read Homogeneous Differential Equation

Linear Differential Equations

A differential equation is considered linear if the dependent variable and its derivatives appear only to the first power and are not multiplied by each other.

The n^th order linear differential equation is of the form:

$a_{0} (x) \frac{d ^{n} y}{d x ^{n}} + a_{1} (x) \frac{d ^{n - 1} y}{d x ^{n - 1}} + \dots\dots .. + a_{n} (x) y = φ (x), w h ere a_{0} (x), a_{1} (x) \dots . a_{n} (x)$ are called the coefficients of the differential equation.

Note that a linear differential equation is always of the first degree. However, not every differential equation of the first degree. e.g. the differential equation $\frac{d ^{2} y}{d x ^{2}} + (\frac{d y}{d x})^{3} + y^{2} = 0$ is not linear, though its degree is 1.

(i) Linear Differential Equations of First Order:

The most general form of a linear differential equation of first order is $\frac{d y}{d x} + P y = Q$ , where $P_{&} Q$

are functions of x.

To solve such an equation multiply both sides by $e^{\int P d x}$ .

So that we get $e^{\int P d x} [\frac{d y}{d x} + P y] = Q e^{\int P d x}$ ...(i)

$\frac{d}{d x} (e^{\int P d x} \cdot y) = Q e^{\int P d x}$ ...(ii)

On integrating equation (ii), we get

$y e^{\int P d x} = \int Q e^{\int P d x} d x + c$

This is the required general solution.

Also Read: First Order Differential Equation

(ii) Equation Reducible to Linear Form:

The equation of the form $\frac{d y}{d x} + P y = Q y^{n}$ , where P and Q are functions of x, is called Bernoulli’s equation.

On dividing by $y^{n}$ , we get $y^{- n} \frac{d y}{d x} + P y^{- n + 1} = Q$

Let $y^{- n + 1} = t, so t ha t (- n + 1) y^{- n} \frac{d y}{d x} = \frac{d t}{d x}$

then equation becomes $\frac{d t}{d x} + P (1 - n) t = Q (1 - n)$

which is linear with t as a dependent variable.

3.0Solved Examples of Differential Equations

Example 1: Find the order and degree of the following differential equation: $\frac{d ^{2} y}{d x ^{2}} = sin (\frac{d y}{d x})$

Solution: The given differential equation has order 2. Since the given differential equation is not a polynomial in the differential coefficients, the degree of the equation is not defined.

Example 2: Solve the differential equation $x y \frac{d y}{d x} = \frac{1 + y ^{2}}{1 + x ^{2}} (1 + x + x^{2})$

Solution: Differential equation can be rewritten as

$x y = (1 + y^{2}) (1 + \frac{x}{1 + x ^{2}}) \Rightarrow \frac{y}{1 + y ^{2}} d y = (\frac{1}{x} + \frac{1}{1 + x ^{2}}) d x$

Integrating, we get

$\frac{1}{2} ln (1 + y^{2}) = ln x + tan^{- 1} x + ln c \Rightarrow 1 + y^{2} = c x e^{t a n^{- 1} x}$

Example 3: Solve $\frac{d y}{d x} = \frac{x - 2 y + 1}{2 x + 2 y + 3}$

Solution:

$\frac{d y}{d x} = \frac{x - 2 y + 1}{2 x + 2 y + 3}$

$⊨^{2 x d y + 2 y d y + 3 d y} = x d x - 2 y d x + d x$

$\Rightarrow (2 y + 3) d y = (x + 1) d x - 2 (x d y + y d x)$

On integrating, we get

$\Rightarrow \int (2 y + 3) d y = \int (x + 1) d x - \int 2 d (x y)$

Solving : $y^{2} + 3 y = \frac{x ^{2}}{2} + x - 2 x y + c$

Example 4: Find the degree of homogeneity of function

(i) $f (x, y) = x^{2} + y^{2}$

(ii) $f (x, y) = (x^{\frac{3}{2}} + y^{\frac{3}{2}}) / (x + y)$

(iii) $f (x, y) = sin (\frac{x}{y})$

Solution:

(i) $f (λ x, λ y) = λ^{2} x^{2} + λ^{2} y^{2} = λ^{2} (x^{2} + y^{2}) = λ^{2} f (x, y)$

degree of homogeneity ⟶ 2

(ii) $f (λ x, λ y) = \frac{λ ^{3/2} x ^{3/2} + λ ^{3/2} y ^{3/2}}{λ x + λ y} f (λ x, λ y) = λ^{\frac{1}{2}} f (x, y)$

degree of homogeneity ⟶ 1/2

(iii) $f (λ x, λ y) = sin (\frac{λ x}{λ y}) = λ^{\circ} sin (\frac{x}{y}) = λ^{\circ} f (x, y)$

degree of homogeneity ⟶ 0

Example 5: The solution of the given differential equation $\frac{d y}{d x} = \frac{s i n y + x}{s i n 2 y - x c o s y} s$ -

(A) $sin^{2} y = x sin y + \frac{x ^{2}}{2} + c$ (B) $sin^{2} y = x sin y - x + c$

(C) $sin^{2} y = x + sin y + x^{2} + c$ (D) $sin^{2} y = x - sin y + \frac{x ^{2}}{2} + c$

Ans. (A)

Solution:

Here, $\frac{d y}{d x} = \frac{s i n y + x}{s i n 2 y - x c o s y}$

$\Rightarrow (cos y) \frac{d y}{d x} = \frac{s i n y + x}{2 s i n y - x} (p u t sin y = t)$

$\Rightarrow \frac{d t}{d x} = \frac{t + x}{2 t - x}$ (put t=v x)

$\frac{x d v}{d x} + v = \frac{v + 1}{2 v - 1}$

$∴ \frac{x d v}{d x} = \frac{v + 1}{2 v - 1} - v = \frac{v + 1 - 2 v ^{2} + v}{2 v - 1}$

or $\frac{2 v - 1}{- 2 v ^{2} + 2 v + 1} d v = \frac{d x}{x}$

on solving, we get

$sin^{2} y = x sin y + + c$ .

Example 6: Solve $\frac{d y}{d x} = \frac{x + 2 y + 3}{2 x + 3 y + 4}$

Solution:

Equation reducible to homogeneous form

Put $x = X + h, y = Y + k 0$

We have $\frac{d Y}{d X} = \frac{X + 2 Y + ( h + 2 k + 3 )}{2 X + 3 Y + ( 2 h + 3 k + 4 )}$

To determine h and k, we write

$h + 2 k + 3 = 0, 2 h + 3 k + 4 = 0 \Rightarrow h = 1, k = - 2$

So that $\frac{d Y}{d X} = \frac{X + 2 Y}{2 X + 3 Y}$

Putting $Y = V X$ , we get

$V + X \frac{d V}{d X} = \frac{1 + 2 V}{2 + 3 V} \Rightarrow \frac{2 + 3 V}{3 V ^{2} - 1} d V = - (\frac{d X}{X})$

$\Rightarrow [\frac{2 + 3}{2 ( 3 V - 1 )} - \frac{2 - 3}{2 ( 3 V + 1 )}] d V = - (\frac{d X}{X})$

$\Rightarrow \frac{2 + 3}{2 3} lo g (3 V - 1) - \frac{2 - 3}{2 3} lo g (3 V + 1) = (- lo g X + c)$

$∣^{\frac{2 + 3}{2 3}} lo g (3 Y - X) - \frac{2 - 3}{2 3} lo g (3 Y + X) = A$

where A is another

constant and

$X = x - 1, Y = y + 2 .$

Example 7: Which of the following equations is a linear differential equation?

(A) $\frac{d y}{d x} + x y^{2} = 1$ (B) $x^{2} + y = e^{x}$

(C) $\frac{d y}{d x} + 3 y = x y^{2}$ (D) $x \frac{d y}{d x} + y^{2} = sin x$

Ans. (B)

Solution:

Clearly answer is (B)

Example 8: Solve the differential equation $\frac{d y}{d x} + \frac{y}{x} = x^{2} y^{6}$ .

Solution:

The provided differential equation can be written as $\frac{1}{y ^{6}} \frac{d y}{d x} + \frac{1}{x y ^{5}} = x^{2}$

Putting y^{-5}=v so that

$- 5 y^{- 6} \frac{d y}{d x} = \frac{d v}{d x} or y^{- 6} \frac{d y}{d x} = - \frac{1}{5} \frac{d v}{d x}$

we get

$- \frac{1}{5} \frac{d v}{d x} + \frac{1}{x} v = x^{2} \Rightarrow \frac{d v}{d x} - \frac{5}{x} v = - 5 x^{2}$ ...(i)

This is the standard form of the linear differential equation having integrating factor

I.F = $e^{\int - \frac{5}{x} d x} = e^{- 5 l o g x} = \frac{1}{x ^{5}}$

Multiplying both sides of (i) by I.F. and integrating w.r.t. x

We get $v. \frac{1}{x ^{5}} = \int - 5 x^{2} \cdot \frac{1}{x ^{5}} d x$

$\Rightarrow \frac{v}{x ^{5}} = \frac{5}{2} x^{- 2} + c$

$\Rightarrow y^{- 5} x^{- 5} = \frac{5}{2} x^{- 2} + c$

which is the required solution.

Also Read Differential Equation previous year questions with solutions

4.0Practice Question on Differential Equations

1. Find the order and degree of differential equation $[\frac{d ^{2} x}{d t ^{2}}]^{3} + [\frac{d x}{d t}]^{4} - x t = 0$

(A) order : 3 ; degree : 2 (B) order : 2 ; degree : 4

(C) order : 2 ; degree : 3 (D) order : 4 ; degree : 2

2. Obtain the differential equation of the family of circles $x^{2} + y^{2} + 2 gx + 2 f y + c = 0$ ; where $g, f & c$

are arbitrary constants.

(A) $[1 + (y^{'})^{2}] \cdot y^{'''} - 3 y^{'} (y^{''})^{2} = 0$

(B) $[1 + (y^{'})^{2}] \cdot y^{'''} + (y^{''} + 2 y^{'}) (y^{''})^{2} = 0$

(C) $[1 + (y^{'})^{2}] \cdot y^{'''} + 3 y^{'} (y^{''})^{2} = 0$

(D) $[1 - (y^{'})^{2}] \cdot y^{'''} + 3 y^{'} (y^{''})^{2} = 0$

3. Solution of the differential equation $\frac{d y}{d x} = sin (x + y) + cos (x + y)$ is

(A) $tan (\frac{x + y}{2}) = a e^{x} - 1$

(B) $tan (\frac{x - y}{2}) = a e^{x} - 1$

(C) $tan (\frac{x + y}{2}) = a e^{- x} - 1$

(D) $tan (\frac{x + y}{2}) = a e^{x} + 1$

4. Solution of the differential equation $\frac{x d x + y d y}{x ^{2} + y ^{2}} = \frac{x d y - y d x}{x ^{2}}$ is

(A) $x^{2} - y^{2} = \frac{y}{x} + c$

(B) $x^{2} + y^{2} = - \frac{y}{x} + c$

(C) $x^{2} + y^{2} = \frac{x}{y} + c$

(D) $x^{2} + y^{2} = \frac{y}{x} + c$

Answer Sheet:

Question	1	2	3	4
Answer	C	C	A	D

Also Read

Divergence and Curl	Differentiability	Applications of Derivatives
Divergence Theorem	Differential Calculus	Partial Differential Equations

1.0Learning Outcomes

After completing this chapter, you will be able to:

Understand the concept of differential equations.
Identify the order and degree of a differential equation.
Form differential equations from given functions.
Solve first-order first-degree differential equations.
Apply the method of variable separation.
Verify the solution of differential equations.
Solve JEE Main, JEE Advanced, and CBSE Board questions effectively.

2.05-Minute Quick Revision

Must-Revise Concepts

Differential Equation
Order of Differential Equation
Degree of Differential Equation
Formation of Differential Equations
General Solution
Particular Solution
Variable Separable Method
Homogeneous Differential Equations
Verification of Solutions
Applications of Differential Equations

Important Formula Recall

General form: $F (x, y, \frac{d y}{d x}) = 0$
Variable separable form: $\frac{d y}{d x} = f (x) g (y)$
Separated form: $\frac{d y}{g ( y )} = f (x), d x$

Important Properties / Shortcut Tips

Determine the order before solving the equation.
Simplify the equation before identifying its degree.
Check whether variables can be separated.
Always include the constant of integration.
Verify the obtained solution by differentiation whenever required.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Order & Degree	High	Easy
Formation of Differential Equations	High	Moderate
Variable Separable Method	Very High	Moderate
General & Particular Solutions	High	Moderate
Homogeneous Differential Equations	Very High	Moderate–High
Verification of Solutions	Moderate	Easy–Moderate
Mixed Differential Equation Problems	Very High	High

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

Common Mistake	Correct Approach
Confusing order with degree	Order depends on the highest derivative, while degree depends on its power after simplification.
Forgetting to separate variables correctly	Rearrange the equation completely before integrating.
Omitting the constant of integration	Always include + C in the general solution.
Incorrect integration	Apply the appropriate integration formula carefully.
Ignoring initial conditions	Use the given conditions to determine the particular solution.
Not verifying the final answer	Differentiate the solution and substitute it back into the original equation.
Algebraic simplification errors	Simplify expressions before and after integration.

6.0ALLEN Faculty Tips

Strengthen your integration concepts before studying this chapter.
Memorize the definitions of order and degree thoroughly.
Practice variable separable equations regularly.
Solve formation-based questions to improve conceptual understanding.
Verify your answers whenever possible.
Practice Previous Years' Questions (PYQs) for better speed and accuracy.

7.0PYQ Trend Analysis

Differential Equations is a frequently tested chapter in CBSE Board, JEE Main, and JEE Advanced. Questions generally focus on identifying order and degree, solving first-order differential equations, and applying the method of separation of variables.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Order & Degree	1	Easy
Formation of Differential Equations	1	Moderate
Variable Separable Equations	2–3	Moderate
General & Particular Solutions	1	Moderate
Mixed Differential Equation Problems	1	Moderate–High

8.0Most Common Question Types

Finding the order and degree of a differential equation.
Forming differential equations from given functions.
Solving variable separable differential equations.
Finding general and particular solutions.
Verifying whether a given function satisfies a differential equation.
Applying integration techniques to solve differential equations.
Solving concept-based and previous years' examination questions.

9.0Smart Revision Strategy

Revise integration formulas before starting this chapter.
Learn the concepts of order and degree thoroughly.
Practice formation of differential equations.
Solve variable separable equations.
Revise general and particular solutions.
Verify solved equations to improve accuracy.
Practice Previous Years' Questions (PYQs).
Complete revision with a timed chapter test.

Test your Knowledge

question 1 of 5

The order of a differential equation is the:

1.Number of terms in the equation

2.Number of variables

3.Highest order derivative present

4.Degree of the equation

Class 12 Maths Chapter 9: Differential Equations

1.0Chapter Snapshot

2.0Chapter 9 Smart Study Material by ALLEN Experts: Download PDFs

Test your Knowledge

question 1 of 5

The order of a differential equation is the:

Frequently Asked Questions

Where are differential equations used in real life?

Why is the Differential Equations chapter important for JEE Main and JEE Advanced?

What is a differential equation in Class 12 Mathematics?

What is the difference between the order and degree of a differential equation?

What is the method of separation of variables?

Join ALLEN!

Class 12 Maths Chapter 9: Differential Equations

1.0Chapter Snapshot

2.0Chapter 9 Smart Study Material by ALLEN Experts: Download PDFs

Differential Equations

1.0Order Of Differential Equations

2.0Degree of Differential Equations

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties / Shortcut Tips

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

Frequently Asked Questions

Where are differential equations used in real life?

Why is the Differential Equations chapter important for JEE Main and JEE Advanced?

What is a differential equation in Class 12 Mathematics?

What is the difference between the order and degree of a differential equation?

What is the method of separation of variables?

Join ALLEN!

Differential Equations

1.0Order Of Differential Equations

2.0Degree of Differential Equations

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties / Shortcut Tips

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

Test your Knowledge

question 1 of 5

The order of a differential equation is the: