CBSE Class 12 Mathematics Notes Chapter 9 Differential Equations

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.

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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.

Order of Differential Equations

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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’’.

Degree of Differential Equations

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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

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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

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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

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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

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