Lesson Plan Math Class XII Ch- 9 | Differential Equations

LESSON PLAN MATHEMATICS
CLASS - XII
CHAPTER - 9 : DIFFERENTIAL EQUATIONS

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. 

Rmb dav centnary public school Nawanshahr
NAME OF THE TEACHER				DINESH KUMAR
CLASS	10+2	CHAPTER	09	SUBJECT	MATHEMATICS
TOPIC	DIFFERENTIAL EQUATIONS			DURATION : 20 Class Meetings

PRE- REQUISITE KNOWLEDGE:

• Knowledge of Trigonometry chapter 3 Class XII

• Knowledge of inverse trigonometric functions chapter 2 class XII

• Knowledge of Differentiation chapter 5 class XII

• Knowledge of Integration chapter 7 class XII

TEACHING AIDS:
Green Board, Chalk,  Duster, Charts, smart board, projector, laptop etc.
METHODOLOGY:   Lecture method
LEARNING 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. (Deleted)

• 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.

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.

PROCEDURE

[Click Here for Complete Explanation]

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.

Differential Equation:

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. 

A differential equation is an equation involving an unknown function and its derivatives with respect to one or more variables.

Involves derivatives with respect to a single variable.

Example: $\frac{dy}{dx} + y = 0$

This is an ordinary differential equation which involves the derivative with respect to single variable.

Order of the differential equations:

Order of the highest order derivative is called the order of the differential equation.

Degree of Differential Equation:

If the differential equation is the polynomial equation in derivatives then exponent of the highest order derivative is called its degree.

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.

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

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:  

 

Equations reducible to variable separable form:

A differential equation that cannot be directly separated, but can be converted into separable form using a suitable substitution, is called an equation reducible to variable separable form.
Basic Idea:

We try to transform the given equation into the standard separable form:

Then we separate variables:

Now taking integral on both side and solve this equation.

Homogeneous Differential Equations:

A homogeneous differential equation is a first-order differential equation of the form:

That is, the right-hand side is a function of the ratio    only.

Alternative Definition:

A differential equation is called homogeneous if both the numerator and denominator are homogeneous functions of the same degree.

Example:   

Here, both numerator and denominator are of degree 2 → so it is homogeneous.

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 for solving Homogeneous Differential Equations

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 .

d) Replace   v by y/x in the solution obtained above so as to get the solution in x and y form.

e) 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.

Linear Differential Equations:

A linear differential equation is a differential equation in which the dependent variable and its derivatives appear only in the first degree (power 1) and are not multiplied together.

Standard Form (First Order):

Where:

y = dependent variable

x = independent variable

P(x) , Q(x) = functions of x only or constant 

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

 

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.

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.

Competency based assessment can be taken so as to ensure if the learning outcomes have been achieved or not. e.g.

• Puzzle

• Quiz

• Misconception check

• Peer check

• Students discussion

• Competency Based Assessment link: M C Q

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