Differential Equations For Class 12

Differential Equations Definition

Differential equations class 12 generally tells us how to differentiate a function “f” with respect to an independent variable. A differential equation is in the form of dy/dx = g(x), where y is equal to the function f(x). Differential equations have a variety of applications, in Physics, Chemistry, Biology, Anthropology, Geology, Economics, etc. We are going to study the basic concepts related to differential equations in detail below.


Differential Equations Class 12 Topics

The topics and sub-topics we are going to cover in differential equations class 12 are as follows:

  • Introduction of Differential Equations.

  • Classification of Differential Equations.

  • What is meant by the order of a differential equation?

  • What is meant by the degree of a differential equation


Introduction of Differential Equations

  • Any equation which contains derivatives, either ordinary derivatives or partial derivatives is known as differential equation

  • A differential equation in Mathematics is an equation that relates one or more than one functions and their derivatives.

  • Physical quantities are represented by the functions in differential equations and the derivatives represent their rates of change, and the differential equation defines a relationship between the two. 


Classification of Differential Equations

The differential equations can be classified into the following-

  1. Ordinary Differential Equations.

  2. Partial Differential Equations.

  3. Linear Differential Equations.

  4. Non-linear Differential Equations.

  5. Homogeneous Differential Equations.

  6. Non-homogenous Differential Equations.

However, the Ordinary Differential Equations and the Partial Differential Equations are the most important and common differential Equations.


Ordinary and Partial Differential Equations

  • "Ordinary Differential Equations" also known as (ODEs) have a single independent variable (like y).

  • "Partial Differential Equations" also known as (PDEs) have two or more independent variables.


Order of Differential Equations:

  • The Order of any differential equation can be defined as the highest derivative.

  • The order can be the first derivative, second derivative, etc.


Here are a Few Examples:

Example 1) dydx + y2 = 2x

The above equation has only the first derivative dydx, therefore it is said to be in "First Order".


Example 2) d2ydx2 + xy = cos(x)

This equation has a second derivative d2ydx2, therefore we can say that the differential equation is in "Order 2".


Example 3) d3ydx3 + xdydx + y = ex

This equation has a third derivative d3ydx3 which outranks the dydx, so we can say that the differential equation is of “Order 3”.


Degree of a Differential Equation -

The degree of a differential equation can be defined as the exponent of the highest derivative.

Example 1)

(dy dx)2 + y = 6x2

The highest derivative in the given equation is just dy/dx, and it has an exponent of 2, so we can say that it is "Second Degree".

We can say that the given equation above is First Order Second Degree Ordinary Differential Equation.


Example 2)

d3ydx3 + (dydx)2 + y = 6x2

The highest derivative in the above equation is d3y/dx3, but it has no exponents, so the above equation is "First Degree".


Note: The exponent of 2 on dy/dx does not count, because it is not the highest derivative).


We can say that the given equation above is the Third Order First Degree Ordinary Differential Equation.


General and Particular Solution of a Differential Equation

The general solution of the differential equation is one that contains arbitrary constants.


Whereas a particular solution of the differential equation is defined as a differential equation that is solution free from arbitrary constants obtained from the general solution by giving particular values to the arbitrary constants.


The methods of Solving First Order and First Degree Differential Equations are

Here are the three methods that are used to solve the first-order and the first-degree differential equations. 


Here they are -


Linear differential equations

Differential equations with variables separable.

Homogeneous(same) differential equations


1. What is Linear Differential Equation?

A linear differential equation can be defined as an equation where P(x) and Q(x) are two continuous functions in the domain of validity of the differential equation. 

If P(x) or Q(x) is equal to 0, we can reduce the differential equation to a variable separable form which can be easily solved.


dydx + P(x)y = Q(x)

             

where P(x) and Q(x) are the functions of x.


Applications of Differential Equations

Here are a few applications of Differential Equations -

1) Differential equations generally describe various exponential growths and decays.

2) They are also used to describe the rate of change in investment return over time.

4) We can describe the movement of electricity with the help of it.

5) Differential Equations help economists in finding optimum investment strategies.

6) We can describe the motion of waves or a pendulum using these equations.

7) It is also used in Physics for heat conduction analysis.


Real - Life Examples of the Application of Differential Equation

If the number of rabbits we have is more the more baby rabbits we get. Then these rabbits grow up and have babies too. The population will grow faster.


The important parts of this growth in population are:

  •  population N at any time t

  •  growth rate r

  • Rate of change of population\[\frac{d}{dt}\]N

Now let us assume some actual values:

  • Let the population N =1000

  • Let the growth rate r = 0.01 new rabbits per week for every current rabbit

The population's rate of change is \[\frac{d}{dt}\]N then 1000 × 0.01 which is equal to 10 new rabbits per week.


This is true at a specific time and doesn't show that the population is constantly increasing.


So we can say that the rate of change (at any instant) is the growth rate times the population at that instant:


\[\frac{dN}{dt}\] = rN


And that is a Differential Equation because it has a function which is equal to N(t) and its derivative.


We can say that differential Equations can describe how populations change. They are a natural way to describe how many things change in the universe.


Here’s a graph showing Growth of Population-

(image will be uploaded soon)


Questions to be Solved:

Question 1) Find out the degree and the order of the given Differential Equation:

\[\frac{dy}{dx}\] - sin x = 0

Solution) The highest order derivative present in the given differential equation is \[\frac{dy}{dx}\] , so the differential order is order one. The highest power \[\frac{dy}{dx}\] is raised to is one, so the degree is one.

FAQ (Frequently Asked Questions)

Question 1) What are Differential Equations Used for?

Answer) Differential equations are used to model the behavior of complex systems in biology and economics.

Question 2) What does a Differential Equation Mean?

Answer) An equation which contains derivatives, either ordinary derivatives or partial derivatives are known as differential equations.

Question 3) What are the Types of Differential Equations?

Answer) The different types of differential equations are -

  1. Ordinary Differential Equations.

  2. Partial Differential Equations.

  3. Linear Differential Equations.

  4. Non-linear Differential Equations.

  5. Homogeneous Differential Equations.

  6. Non-homogenous Differential Equations.

Question 4) What are the Two Types of Differential Equations?

Answer) Differential equations can be classified into two types:

  1. Ordinary differential equation - A differential equation that does not involve partial derivatives is known as Ordinary Differential equations.

  2. Partial differential equations- A differential equation that involves partial derivatives is known as partial differential equations.