Differential equations  

What is Differential equations

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.

 

Differential Equation

 

Why Differential Equation

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.

\\\frac{\mathrm{d} T}{\mathrm{d} t}=0.05^{\circ}C \\\\dT=0.05dt \\\\\int_{T}^{27^{\circ}C}dT=0.05\int_{10:00 \;am}^{2:00 \;pm} dt \\\\T-27=0.05\times 240 \\T=12+27=39^{\circ}C

Here you learn the basics of differential equations.

After reading this chapter  you will be able to:

  • Identify the order of a differential equation
  • Distinguish between the general solution and a particular solution of a differential equation
  • Determine the nature of any function
  • Determine the dependency of a physical quantity on another quantity

Important Topics

  • Order and Degree of a Differential equation
  • General and Particular solution
  • Formation of differential equation
  • Methods of solving different types of differential equation

 

Overview of Chapter

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.

 

Types of Solution of Differential equation

General solution

The general solution of a differential equation having nth 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

  1. F be a function depends upon the parameter a and b then it is represented by an equation of form F(x, y, a, b)=0.
  2. Differentiate F with respect to the x we get the second equation G involving \frac{\mathrm{d}y }{\mathrm{d} x},\;y,\;x,\;a,\;b i.e. \text{G}(\frac{\mathrm{d}y }{\mathrm{d} x},\;y,\;x,\;a,\;b)=0.
  3. Differentiate again to eliminate two parameters from the above two equations and we get third equation H involving \frac{\mathrm{d^2}y }{\mathrm{d} x^2},\;\frac{\mathrm{d}y }{\mathrm{d} x},\;y,\;x,\;a,\;b i.e. \text{H}(\frac{\mathrm{d^2}y }{\mathrm{d} x^2},\;\frac{\mathrm{d}y }{\mathrm{d} x},\;y,\;x,\;a,\;b)=0.
  4. The final equation becomes \text{F}(\frac{\mathrm{d^2}y }{\mathrm{d} x^2},\;\frac{\mathrm{d}y }{\mathrm{d} x},\;y,\;x)=0 after eliminating both the parameters.

 

Different types 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 first-order linear differential equation.

 

How to prepare Differential equations

  • To understand this topic you have to go continuity and differentiability where you learn the differential.
  • Start with understanding the basic concepts of Differentiation which play an important role in this chapter.
  • Once you’re clear with basic concepts, move to important or standard formulas used in differentiation.
  • Do more and more questions on different types of differential equations.
  • While going through concept make sure you understand the function and try to solve them by your own, as some times you forget the formula so it will be very helpful for you.
  • Solve all the questions of NCERT book sequentially and then go to previous year papers.
  • Try to learn more and more things from each and every question.
  • At the end of chapter try to make your own formulas to revise quickly before exams or anytime when you required to revise the chapter, it will save lots of time for you.

 

Best books for the preparation of Differential equations:

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.

 

Maths Chapter-wise Notes for Engineering exams

Chapters 

Chapters Name

Chapter 1

Sets, Relations, and Functions 

Chapter 2

Complex Numbers and Quadratic Equations

Chapter 3

Matrices and Determinants

Chapter 4

Permutations and Combinations

Chapter 5

Binomial Theorem and its Simple Applications

Chapter 6

Sequence and Series

Chapter 7

Limit, Continuity, and Differentiability

Chapter 8

Integral Calculus

Chapter 10

Coordinate Geometry

Chapter 11

Three Dimensional Geometry

Chapter 12 

Vector Algebra

Chapter 13

Statistics and Probability

Chapter 14 

Trigonometry

Chapter 15 

Mathematical Reasoning

Chapter 16

Mathematical Induction

Topics from Differential equations

  • Ordinary differential equations, their order and degree ( AEEE, JEE Main, VITEEE, KVPY SA, KVPY SB/SX ) (24 concepts)
  • Formation of differential equations ( AEEE, JEE Main, VITEEE, KVPY SA, KVPY SB/SX ) (9 concepts)
  • Solution of differential equations by the method of separation of variables ( AEEE, JEE Main, VITEEE, KVPY SA, KVPY SB/SX ) (36 concepts)
  • Solution of homogeneous and linear differential equations ( AEEE, JEE Main, VITEEE, KVPY SA, KVPY SB/SX ) (78 concepts)
  • Differential equations ( AEEE, JEE Main, VITEEE, KVPY SA, KVPY SB/SX ) (11 concepts)

Important Books for Differential equations

  • Differential equations Book
  • Differential equations Book
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