The differential equation is the part of the calculus in which an equation defining the unknown function y=f(x) and one or more of its derivatives in it. In mathematics, calculus depends on derivatives and derivative plays an important part in the differential equations. Every year you will get at least 1  2 questions in JEE Main and other exams. From the exam point of view, it is the most important chapter in calculus because solving these equations often provide information about how quantities change and frequently provides insight into how and why the changes occur. This chapter starts with the basics of differential calculus and later it turns into the toughest part of the differential calculus. Differential equations have a remarkable ability to predict the world around us.
Weather Forecasting is the best example of differential calculus as we know that weather forecast depends on many factors like wind speed, moisture level, temperature, etc.
Let suppose today's weather forecast is like that temperature will increase by 0.05 degrees per min. Will you able to determine the temperature at 2:00 pm. If the temperature at 10:00 am will be equal to body temperature (27°C).
How you will solve it?
The total increase in temperature is equal to 0.05 multiplied with total time (4hr x 60 min = 240 mins) = 0.05 x 240 = 12°C. Now you add 27°C and 12°C which results in 39°C.
Now solve it by using the differential equation
It is given that the rate of change in Temperature is 0.05°C.
Here you learn the basics of differential equations.
After reading this chapter you will be able to:
Important Topics
Order of a differential equation
The order of a differential equation is defined as the order of the highest order derivative of the dependent variable concerning the independent variable involved in the given differential equation.
Degree of a Differential equation
The degree of a differential equation is defined as the power of the highest derivative after the equation has been made rational and integral in all of its derivatives.
General solution
The general solution of a differential equation having n^{th} order is defined as the solution having at least n number of arbitrary constant.
Particular solution
The Particular solution of a differential equation is obtained by the general solution which is free from arbitrary constant.
Formation of differential equation
Differential equations with variables separable: It is defined as a function F(x,y) which can be expressed as f(y)dy = g(x)dx, where, g(x) is a function of x and h(y) is a function of y.
Homogeneous differential equations: If a function F(x,y) which can be expressed as f(x,y)dy = g(x,y)dx, where, f and g are homogenous functions having the same degree of x and y.
Linear differential equations: A differential equation of the form y'+Py=Q where P and Q are constants or functions of x only, is known as a firstorder linear differential equation.
First, finish all the concepts, example and questions given in NCERT Maths Book along with Miscellaneous Exercise. You must be thorough with the theory of NCERT. Then you can refer to the book Calculus by Dr. SK goyal or RD Sharma but make sure you follow any one of these not all. Differential equations is explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find the book that best suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.
Chapters 
Chapters Name 
Chapter 1 

Chapter 2 

Chapter 3 

Chapter 4 

Chapter 5 

Chapter 6 

Chapter 7 

Chapter 8 

Chapter 10 

Chapter 11 

Chapter 12 

Chapter 13 

Chapter 14 

Chapter 15 

Chapter 16 
The solution of the differential equation satisfying the condition is
The differential equation of the family of circles with fixed radius 5 units and centre on the line is
The differential equation of all circles passing through the origin and having their centres on the axis is