Differential Equations

DIFFERENTIAL EQUATIONS

 Introduction:
Differential equations are meant to state relationship between a variable (to be specific, a function) and the rate of change of another associated with it. If we write,- Y = f(x) , then we are saying that Y is a function of X. Here x is an independent variable and y is dependent variable. In short, an equation with a derivative of an independent variable along with the dependent variable (or not) and a constant is named after “Differential Equation.”

 Key concepts:

 Order and Degree Of a Differential Equation:
The highest order derivative present in the differential equation is the “order” of the equation. Similarly, the power associated with the highest order of derivative is the “degree” of that equation.
In the given figure- 1,

(I) Order : 2
Degree: 1
(II) Order : 2
Degree: 1
(III) Order : 3
Degree: 1
(IV) Order : 1
Degree: 1
Always remember – “order and degree of a differential equation are positive integers.

 Solution of differential equation:
Here solution means the relationship between the independent and the dependent variable without having derivative inside. It has two specific ways which are going to be explained:

  1. General Solution:
    It contains the same number of the arbitrary constants as the order of the differential equation.

In figure- 2, we can see all of the three solutions have excluded the derivatives and established a relation between variables only.

  1. Particular Solution:
    After getting the general one, just assign the values of the constants and you will get particular solution.
    From figure- 2, if we find the value of C1, C2, then it will become a particular solution.
    Figure-3, gives us a clear idea of particular solution
     Formation of Differential Equation:
    There are some steps which are needed to follow while forming a differential equation.

(i) You need to differentiate the equations as many times as the number of arbitrary constants are present in the equation.
(ii) Eliminate all constants.
(iii) Say, there are n constants. Then, the equation will be of nth order and you will get n equations.

 Method of Solving 1st Order 1st Degree Differential Equation:
In this article, we shall deal with only three methods which are as followed.

(a) Variable Separable Method:
In this method, variables are separable i.e. coefficient of dx is only a function of x and coefficient of dy is only a function of y.
From figure- 4, we can see the procedure.

(b) Homogeneous Differential Equation:
The general form of this equation is,- dy/dx = v + x. dv/dx

Ratio of two homogeneous functions of same degree is to be taken as a single variable (say:”v”) here. Then integration goes on.

(c) Linear Differential Equation:
The general form of this equation is,- dy/dx + Py = Q

Here the coefficient of independent variable takes major part to solve this equation.

In figure – 6, the steps are to be followed and then the equation is prepared.

 Conclusion:
There are more methods but the syllabus comprises to the above only. The article has all the points which are detailed in NCERT like order and degree of differential equations, formation and methods to solve those. It will help you for last minute revision and to summarize your learnings will give you a new and better approach.