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Al Wakrah

Lesson Plan On Differential Equations

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Published in: Mathematics | Maths
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Brief description to Differential equations.

Shaik M / Doha

10 years of teaching experience

Qualification: MSc in Applied Mathematics , Bachelor of Education and Phd in Computational Statistics

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  1. LESSON PLAN Couse Ref: LESSON PLAN Ref : Sub •ect / Course: To ic: Lesson Title: Level: l. Introduction: Focus of Lesson Differential Calculus Mathematics Differential E uation Problem Solvin of Differential Intermediate Lesson 0b •ectives uations 2. Student Learning : What we learned about students' understanding based on data collected 3. Teaching Strategies: What we noticed about our own teaching 4. Strengths & Weaknesses of adopting the Lesson Study process 5.Possesses sound knowledge of different domains and technologies that were part of the teaching curriculum. 6. Provides interestin and livel new exercise material es ciall for air and rou work o o o o o o o o Summa of Tasks / Actions: Introduction and Example First order differential equations with separable variables First order differential equations of Homogeneous nature Equations reducible to the Homogeneous form First Order Linear Differential Equations References Assignment Answer Key Materials/ E ui ment Introduction A differential equation contains one or more terms involving derivatives of one variable (the dependent variable, y) with respect to another variable (the independent variable, x). For example, dy clx
  2. Example Solution dy dx d2y clx2 From these three above equations, we can observe that dx2 dx First order differential equations with separable variables Let the differential equation be of the form =flx, y) Wheref (x, y) denotes a function in x and y. (1) (1) is separable if it can be expressed in the form f(x, y) = N(y) M(x), N(y) are real valued functions of x and y respectively. dy M (x) Then, we have dx NO) Integrating both sides of (2), we get the solution viz. Example Solve Solution dx 3 e x +x2e Y +eX)dx Integrating both sides, we get + c , which is the solution of the above differential equation, where c is an arbitrary constant.
  3. First order differential e uations of Hom eous nature dy Suppose we have the differential equation of the form — = f (x, y) Where f (x, y) is a function of x and y. (1) is of the homogeneous form if it can be written in the form F _ or F . Also the homogeneity can be spotted when all the terms are of the same order. These equations are solved by putting y = vx, where v * v(x), a function of x. Example Solve (x 2 + Solution All terms are of order 2, hence it is a homogeneous equation — dy 2xy dx 2x2v dx — log(l — v2) = log-x — log c So, x2 — cx = O, is the required solution.
  4. E uations reducible to the Homo eneous form dy ax + by + c Equation of the form (aB 44b) can be reduced to a homogenous form by changing the variables x, y to X, Y by equation x = X + h y = Y + k; where h, k are constants to be chosen so as to make the given equation homogenous, we have dy _ d(Y+k) dY dY dX dY dx dX'dx dX dx dx so equation becomes dX AX + BY + C) Let h and k be chosen so as to satisfy the equation ah + bk + c — bc — Bc Ac-aC These give h — Which are meaning full except when aB = Ab dy_ aX+bY dx AX + BY can now be solved by means of the substitution Y = VX Example dy x + 2y + 3 dx 2x+3y+4 Solution Putx=X+h y=Y+k X +2Y + (h + 2k +3) We have dX 2X +3Y + (2h+3k +4) To determine h and k, we have
  5. Putting Y = VX, we get dX 2+3V 2+3V 31' 2-1 2 vfiv-l x 2 v/ÄV+1 x 2 2 2 log(OY X)- A where A is another constant and X = x 2 2 First Order Linear Differential Equations The most general form of this category of differential equations is —+ Py — Q , where P and Q are constants, or functions of x, along or of the form — + PX Q, where P and Q are constants or functions dy of y, along. We use here an integrating factor, namely e and the solution is obtained by dx + c where c is an arbitrary constant. Example dy 2y = cos x dx Solution It is a linear equation so the integrating factor is Hence, y.e e cos xdx + c = x + c is the required solution 5
  6. References: 1. Differential Equations , 3rd ed. McGraw-Hill Ayres, Frank, 1901-1994. 2. Differential Calculus Book by Shanti Narayan Shanti Narayan , 1962. 3. http://wvvw.sosmath.com/calculus/calculus.html 4. http://wvvw.math.utah.edu/—pa/math.html 5. http://wvvw.sparknotes.com/math/ Take Home Tasks: Assignment : Solve the following differential equations. -o (ii) OBJECTIVE QUESTIONS Choose the correct answer/answers in each of the following dy is a differential equation of dx dr2 a) degree 2 (b)degree 3 (c) order 2 (d) order 1 (ii) tan x is (a) homogenous differential equation (c) extended linear differential equation dx (iii) I.F. for y log y —+ x— log y = O is dy (b) linear differential equation (d) variable separation differential equation (a) log x Objective Solutions : (b) log y (ii) tan (iii) b (c) log xy ANSWERS (d) none of these

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