Differential Equation Formula, Examples & Types

In Class 12 Mathematics, a differential equation is an equation that contains one or more functions and has derivatives. It is defined as a differential equation. The rate of change of a function at some point is defined by the derivative of the function. The primary purpose of differential equation Formula is to study solutions that satisfy the properties of equations and solutions.

Differential Equation is of utmost importance in Class 12 Maths as it asks objective questions along with questions of 5 marks in the exam. It has some basic parts from which objective questions are asked every year, which are covered here.

In this article, the formula, definition, types, method of solving questions, degree, order etc. of Differential Equation are described in detail which makes Differential Equation easy.

What is Differential Equation

A differential equation is an equation that relates one or more functions and their derivatives.

in other word,

A differential equation is an equation which contains one or more terms and the derivatives of one variable with respect to the other variable.

dy/dx = g(x), where y = g(x) or,

We can say that “x” is an independent variable and “y” is a dependent variable.

Examples:

  • y’ = sin x + y
  • (d2y/dx2) + y = 3x + 5
  • (d2y/dt2) + (d2x/dt2) = x
  • (d3y/dv3) + v(dy/dv) = 10xy
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Order of Differential Equation

Generally, Order of a differential equation is defined as the order of the highest order derivative of
the dependent variable with respect to the independent variable involved in the given
differential equation.

For Examples

1. dy/dx = ex , The order of the equation is 1.

2. (d2y/dx2) + y = 0, The order is 2

3. (d3y/dx3) + x2(d2y/dx2) = 0, The order is 3

4. (d4y/dx4) + tanx. y = 0, The order is 4

In above differential equation examples, the highest derivative are of first, second, third and fourth order respectively.

Degree of Differential Equation

If a differential equation is expressible in a polynomial form, then the power of the highest order derivative is called the degree of the differential equation.

Example:

dy/dx + 3x = 0, Degree is 1

  • (d3y/dx3)3 + 3 (d2y/dx2) + 6 (dy/dx) – 12 = 0, Degree is 3
  • (d2y/dx2) + sin(dy/dx) = 0,  It is not a polynomial equation in y′ and the degree of this differential equation can not be defined.
  • (d4y/dx4)2 + (d3y/dx3)3 + 3 (d2y/dx2) + 5 (dy/dx) – 2x = 0, Degree is 2.

Note:

First Order Differential Equation
dy/dx = f(x, y) = y’
Second-Order Differential Equation
d/dx(dy/dx) = d2y/dx2 = f”(x) = y”
Order and degree (if defined) of a differential equation are always positive integers

Types of Differential Equations

Differential equations can be divided into several types. But here according to class 12 maths, we will study about the major three forms from which questions come in the exam.

  • Separation of the variable
  • Homogenous
  • Linear

Separation of the variable

This method i.e. Separation of the variable is the process of solving questions with Differential Equation which is studied in class 12th.

The separation of the variables is done when the differential equation can be written as dy/dx = f(y)g(x) where f is a function of y only and g is a function of x only. Taking a starting position, rewrite this problem as 1/f(y)dy= g(x)dx and then integrate on both sides.

Example:

x dy = (2x2 + 1) dx

dy/dx + y = 1

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Homogenous

A differential equation in which the degree of all the terms is the same is known as a homogenous differential equation.

The Homogenous differential equation can be written as P(x,y)dx + Q(x,y)dy = 0, where P(x,y) and Q(x,y) are homogeneous functions of the same degree. See more examples here:

  • (x – y) y’ = x + 2y
  • y’ = (x + y) / x

Linear Differential Equation

A linear differential equation is defined by the linear polynomial equation, which consists of derivatives of several variables.

The Linear differential equation can be written as dy/dx + Py = Q. where P and Q are either constants or functions of y (independent variable) only.

Steps involved to solve first order linear differential equation:

  • Write the given differential equation in the form dy / dx + Py= Q, where P, Q are constants or functions of x only.
  • Find the Integrating Factor (I.F) = epdx
  • Write the solution of the given differential equation as y (I.F) = (Q × I.F)dx + C

In class XII only three types of questions are asked from Differential Equation. Here the formula, example, process etc. of all the three Differential Equations have been included which is necessary.

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