Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

Lesson Plan Math Class XII Ch- 9 | Differential Equations


E- LESSON PLAN   SUBJECT MATHEMATICS    CLASS 10+2

Lesson Plan, Class XII Subject Mathematics, chapter 9, Differential Equations , for Mathematics Teacher. Effective way of Teaching Mathematics. Top planning by the teacher for effective teaching in the class. E lesson planning for mathematics.

Board – CBSE

CLASS –XII

SUBJECT- MATHEMATICS

CHAPTER 9  :- Differential Equations


TOPIC:- CHAPTER  9 : Differential Equations

DURATION:-  

This chapter is divided into 10 modules and is completed in 20 class meetings.

PRE- REQUISITE KNOWLEDGE:-


TEACHING AIDS:- 

Green Board, Chalk,  Duster, Charts, smart board, projector, laptop etc.

METHODOLOGY:-   Lecture method

OBJECTIVES:-

  • Definition of Differential equations.
  • Order and Degree of differential Equations.
  • General and Particular solutions of Differential equations.
  • Formation of differential equations whose general solution is given.
  • Solution of differential equations by the method of separating the variables.
  • Solution of homogeneous differential equations of first order and first degree.
  • Solutions of Linear Differential equations of the following type  where P and Q are the functions of x  or constant  , where P and Q are the functions of y or constant.
  • Applications of Differential Equations in different field of mathematics.
PROCEDURE :-

Start the session by asking the questions related to the trigonometry, inverse trigonometric functions differentiation and integrations. Now introduce the topic Differential equations  step by step as follows.

S.No

Explanation of the Topics   

 

1

Differential Equations : 

An equation which involves the  independent variable, dependent variable and differential coefficients of dependent variable with respect to the independent variable is called a differential equation. In dy/dx, the dependent variable is y and independent variable is x.

2

Order of the differential equations:

It is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable.

3

Degree of Differential Equation:

Degree of a differential equation is defined if it is the polynomial equation in derivatives. Highest power of the highest order derivative is called its degree. Before finding the degree, the differential equation should be free made from radicals and fractions.  Order and degree(If defined) are always positive integers.

4

General Solution : The solution which contains the arbitrary constants is called general solution.

Particular Solution : The solution which is free from arbitrary constants is called a particular solution.

After giving the knowledge of types of solutions teacher should explain the method of finding the solutions and help the students in solving the problems given in the NCERT book.

5

Now explain the method of formation of differential equation whose  general solution is given by taking some problems from the text book.

6

Now explain the method of solving the first order, first degree differential equations.

These equations are of three types

i) Differential equations in which variables can be separable.
ii) Homogeneous Differential Equations.
iii) Linear Differential Equations.

First of all teacher should explain the method of solving the equations in which variables can be separable.
When the differential equation is of the form f(x)dx = g(y)dy, then variables can be separable. Such equations cab solved by integrating on both sides and the solution is of the form:  

7

Equations reducible to variable separable form:

Differential equation of the form dy/dx = f(ax + by + c) can be reduced to variable separable form by putting ax + by + c = v

Teacher should explain this method and problems related to it.

8

Homogeneous Differential Equations:

If a first order first degree differential equation is expressible in the form

  

Where f(x, y) and g(x, y) are homogeneous functions of the same degree, then it is called a homogeneous differential equation.

Such types of equations can be reduced to the variable separable form by the substitution y = vx
Teacher should explain the complete method of solving such equations by solving some problems.
Algorithm

a) Put y = vx, and dy/dx = v + x dv/dx
b) Shift v to the RHS and separate the variables v and x
c) Integrating on both sides to obtain the solution in terms of v and   x .
Replace   v by y/x in the solution obtained above so as to get the solution in x and y form.
Sometimes a homogeneous differential equation is expressible in the form 

Where f(x, y) and g(x, y) are homogeneous functions of the same degree

In such a situation we substitute x = vy and dx/dy = v + ydv/dy

9

Linear Differential Equations:

A differential equation is linear if the dependent variable (y) and its derivative appear only in first degree. General form of Linear Differential equation is 

 
where P and Q are either the functions of x only or constants. Coefficient of dy/dx should be unity. To solve these equations teacher should explain the following algorithm to the students.

Algorithm

a) Write the differential equation in the standard form dy/dx + Py = Q and then find P and Q

b) Find integrating factor(IF) given by 

c) Multiply on both side of the equation by I.F.

c) Integrating on both side with respect to x so that we get the solution in the form 

Sometimes the L.D.E. is of the form 

 
where P and Q are either the functions of y only or constants. Here

 

and the solution becomes

10

Teacher should explain some applications of differential equations in different fields of mathematics like growth and decay, Newton’s law of cooling, coordinate geometry etc


EXPECTED OUTCOMES:-
  • After studying this lesson students should know the 
  • Differential equations their order , degree , arbitrary constants, general solutions and particular solutions. 
  • Students should also know the different methods of solving the differential equations like: variable separable method, homogeneous differential equation and linear differential equation.
STUDENTS DELIVERABLES:-
  • Review questions given by the teacher. 
  • Students can prepare a power point presentation on basic concepts and different methods of solving the differential equations.  
  • Solve NCERT problems with examples.
EXTENDED LEARNING:-

Students can extend their learning in Mathematics through the RESOURCE CENTRE.  Students can also find many interesting topics on mathematics at cbsemathematics.com

ASSESSMENT TECHNIQUES:-
  • Assignment sheet will be given as home work at the end of the topic. 
  • Separate sheets which will include questions of logical thinking and Higher order thinking skills will be given to the above average students.
  • Class Test , Oral Test , worksheet and Assignments. can be made the part of assessment.
  • Re-test(s) will be conducted on the basis of the performance of the students in the test.
THANKS FOR YOUR VISIT
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