m1 unit ii-jntuworld

MATHEMATICS-IMATHEMATICS-I

CONTENTSCONTENTS Ordinary Differential Equations of First Order and First DegreeOrdinary Differential Equations of First Order and First Degree Linear Differential Equations of Second and Higher OrderLinear Differential Equations of Second and Higher Order Mean Value TheoremsMean Value Theorems Functions of Several VariablesFunctions of Several Variables Curvature, Evolutes and EnvelopesCurvature, Evolutes and Envelopes Curve TracingCurve Tracing Applications of IntegrationApplications of Integration Multiple IntegralsMultiple Integrals Series and SequencesSeries and Sequences Vector Differentiation and Vector OperatorsVector Differentiation and Vector Operators Vector IntegrationVector Integration Vector Integral TheoremsVector Integral Theorems Laplace transformsLaplace transforms

TEXT BOOKSTEXT BOOKS A text book of Engineering Mathematics, Vol-I A text book of Engineering Mathematics, Vol-I

T.K.V.Iyengar, B.Krishna Gandhi and Others, T.K.V.Iyengar, B.Krishna Gandhi and Others, S.Chand & CompanyS.Chand & Company

A text book of Engineering Mathematics, A text book of Engineering Mathematics, C.Sankaraiah, V.G.S.Book LinksC.Sankaraiah, V.G.S.Book Links

A text book of Engineering Mathematics, Shahnaz A A text book of Engineering Mathematics, Shahnaz A Bathul, Right PublishersBathul, Right Publishers

A text book of Engineering Mathematics, A text book of Engineering Mathematics, P.Nageshwara Rao, Y.Narasimhulu & N.Prabhakar P.Nageshwara Rao, Y.Narasimhulu & N.Prabhakar Rao, Deepthi PublicationsRao, Deepthi Publications

REFERENCESREFERENCES

A text book of Engineering Mathematics, A text book of Engineering Mathematics, B.V.Raman, Tata Mc Graw HillB.V.Raman, Tata Mc Graw Hill

Advanced Engineering Mathematics, Irvin Advanced Engineering Mathematics, Irvin Kreyszig, Wiley India Pvt. Ltd.Kreyszig, Wiley India Pvt. Ltd.

A text Book of Engineering Mathematics, A text Book of Engineering Mathematics, Thamson Book collectionThamson Book collection

UNIT - IIUNIT - II

LINEAR DIFFERENTIAL LINEAR DIFFERENTIAL EQUATIONS OF SECOND AND EQUATIONS OF SECOND AND

HIGHER ORDERHIGHER ORDER

UNIT HEADERUNIT HEADER

Name of the Course: B. TechName of the Course: B. TechCode No:07A1BS02Code No:07A1BS02Year/Branch: I Year Year/Branch: I Year

CSE,IT,ECE,EEE,ME,CIVIL,AEROCSE,IT,ECE,EEE,ME,CIVIL,AEROUnit No: IIUnit No: II

No. of slides:18No. of slides:18

S. S. No.No.

ModuleModule LectureLectureNo. No.

PPT Slide PPT Slide No.No.

11 Introduction, Introduction, Complementary functionComplementary function

L1-5L1-5 8-138-13

22 Particular IntegralsParticular Integrals L6-11L6-11 14-1914-19

33 Cauchy’s, Legendre’s Cauchy’s, Legendre’s linear equations, linear equations, Variation of ParametersVariation of Parameters

L12-14L12-14 20-2220-22

UNIT INDEXUNIT INDEXUNIT-II UNIT-II

Lecture-1Lecture-1 INTRODUCTION INTRODUCTION

An equation of the formAn equation of the form DDnny + ky + k11 D Dn-1n-1y +....+ ky +....+ knny = Xy = X Where kWhere k11,…..,k,…..,knn are real constants and are real constants and X is a X is a

continuous function of continuous function of xx is called an ordinary linear is called an ordinary linear equation of order equation of order nn with constant coefficients. with constant coefficients.

Its complete solution isIts complete solution is y y = C.F + P.I= C.F + P.I where C.F is a Complementary Function and where C.F is a Complementary Function and P.I is a Particular Integral.P.I is a Particular Integral. ExampleExample:d:d22y/dxy/dx22+3dy/dx+4y=sinx+3dy/dx+4y=sinx

COMPLEMENTARY FUNCTIONCOMPLEMENTARY FUNCTION

If roots are real and distinct thenIf roots are real and distinct then C.F = C.F = cc1 1 eem1x m1x + …+ c+ …+ ckk e emkxmkx

Example Example 1: Roots of an auxiliary equation are 1,2,3 then 1: Roots of an auxiliary equation are 1,2,3 then C.F = cC.F = c11 e ex x + c+ c22 e e2x 2x + c+ c33 e e3x3x

ExampleExample 2: For a differential equation 2: For a differential equation (D-1)(D+1)y=0, roots are -1 and 1. Hence(D-1)(D+1)y=0, roots are -1 and 1. Hence C.F = cC.F = c1 1 ee-x-x +c +c22 e exx

Lecture-2Lecture-2COMPLEMENTARY FUNCTIONCOMPLEMENTARY FUNCTION

If roots are real and equal thenIf roots are real and equal then C.F = (C.F = (cc11+ c+ c22x +….+ cx +….+ ckkxxkk))eemxmx

Example Example 1: The roots of a differential equation 1: The roots of a differential equation (D-1)(D-1)33y=0 are 1,1,1. Hence y=0 are 1,1,1. Hence C.F.= (cC.F.= (c11+c+c22x+cx+c33xx22)e)exx

ExampleExample 2: The roots of a differential equation 2: The roots of a differential equation (D+1)(D+1)22y=0 are -1,-1. Hence y=0 are -1,-1. Hence C.F=(cC.F=(c11+c+c22x)ex)e-x -x


If two roots are real and equal and rest are real If two roots are real and equal and rest are real and different then and different then C.F=(cC.F=(c11+c+c22x)ex)em1xm1x+c+c33eem3xm3x+….+….

Example : The roots of a differential equation Example : The roots of a differential equation (D-2)(D-2)22(D+1)y=0 are 2,2,-1. Hence (D+1)y=0 are 2,2,-1. Hence C.F.=(cC.F.=(c11+c+c22x)ex)e2x2x+c+c33ee-X-X


If roots of Auxiliary equation are complex say If roots of Auxiliary equation are complex say p+iq and p-iq then p+iq and p-iq then C.F=eC.F=epxpx(c(c11 cosqx+c cosqx+c22 sinqx) sinqx)

Example: The roots of a differential equation Example: The roots of a differential equation (D(D22+1)y=0 are 0+i(1) and 0-i(1). Hence +1)y=0 are 0+i(1) and 0-i(1). Hence C.F=eC.F=e0x0x(c(c11cosx+ccosx+c22sinx) sinx)

=(c=(c11cosx+ccosx+c22sinx)sinx)


A pair of conjugate complex roots say p+iq A pair of conjugate complex roots say p+iq and p-iq are repeated twice then and p-iq are repeated twice then C.F=eC.F=epxpx((c((c11+c+c22x)cosqx+(cx)cosqx+(c33+c+c44x)sinqx)x)sinqx)

ExampleExample: The roots of a differential equation : The roots of a differential equation (D(D22-D+1)-D+1)22y=0 are ½+i(1.7/2) and ½-i(1.7/2) y=0 are ½+i(1.7/2) and ½-i(1.7/2) repeated twice. Hence repeated twice. Hence C.F=eC.F=e1/2x1/2x(c(c11+c+c22x)cos(1.7/2)x+ x)cos(1.7/2)x+ (c(c33+c+c44x)sin(1.7/2)xx)sin(1.7/2)x

Lecture-6Lecture-6PARTICULAR INTEGRALPARTICULAR INTEGRAL

When X = eWhen X = eaxax put D = a in Particular Integral. put D = a in Particular Integral. If If f(a)f(a) ≠ 0 then P.I. will be calculated directly. ≠ 0 then P.I. will be calculated directly. If If f(a)f(a) = 0 then multiply P.I. by = 0 then multiply P.I. by x x and and differentiate denominator. Again put differentiate denominator. Again put D = aD = a.Repeat the same process..Repeat the same process.

ExampleExample 1:y″+5y′+6y=e 1:y″+5y′+6y=exx. Here P.I=e. Here P.I=exx/12/12 Example Example 2:4D2:4D22y+4Dy-3y=ey+4Dy-3y=e2x2x.Here P.I=e.Here P.I=e2x2x/21/21


When X = Sinax or Cosax or Sin(ax+b) or When X = Sinax or Cosax or Sin(ax+b) or Cos(ax+b)Cos(ax+b) then put Dthen put D22= - a= - a22 in Particular in Particular Integral.Integral.

ExampleExample 1: D 1: D22y-3Dy+2y=Cos3x. Here y-3Dy+2y=Cos3x. Here P.I=(9Sin3x+7Cos3x)/130P.I=(9Sin3x+7Cos3x)/130

Example Example 2: (D2: (D22+D+1)y=Sin2x. Here P.I= +D+1)y=Sin2x. Here P.I= -1/13(2Cos2x+3Sin2x)-1/13(2Cos2x+3Sin2x)


When X = xWhen X = xkk or in the form of polynomial then or in the form of polynomial then convertconvert f(D) into the form of binomial f(D) into the form of binomial expansion from which we can obtain Particular expansion from which we can obtain Particular Integral.Integral.

ExampleExample 1: (D 1: (D22+D+1)y=x+D+1)y=x33.Here P.I=x.Here P.I=x33-3x-3x22+6+6 ExampleExample 2: (D 2: (D22+D)y=x+D)y=x22+2x+4. Here +2x+4. Here

P.I=xP.I=x33/3+4x/3+4x


WhenWhen X = eX = eaxaxv then put D = D+a and take out v then put D = D+a and take out eeaxax to the left of f(D). Now using previous to the left of f(D). Now using previous methods we can obtain Particular Integral.methods we can obtain Particular Integral.

ExampleExample 1:(D 1:(D44-1)y=e-1)y=ex x Cosx. Here Cosx. Here P.I=-eP.I=-exxCosx/5Cosx/5

ExampleExample 2: (D 2: (D22-3D+2)y=xe-3D+2)y=xe3x3x+Sin2x. Here +Sin2x. Here P.I=eP.I=e3x3x/2(x-3/2)+1/20(3Cos2x-Sin2x)/2(x-3/2)+1/20(3Cos2x-Sin2x)


When X = x.v then When X = x.v then P.I = [{x – fP.I = [{x – f ""(D)/f(D)}/f(D)]v(D)/f(D)}/f(D)]v Example Example 1: (D1: (D22+2D+1)y=x Cosx. Here +2D+1)y=x Cosx. Here

P.I=x/2Sinx+1/2(Cosx-Sinx)P.I=x/2Sinx+1/2(Cosx-Sinx) ExampleExample 2: (D 2: (D22+3D+2)y=x e+3D+2)y=x ex x Sinx. Here Sinx. Here

P.I=eP.I=exx[x/10(Sinx-Cosx)-1/25Sinx+Cosx/10][x/10(Sinx-Cosx)-1/25Sinx+Cosx/10]


WhenWhen X X is any other function then Particular is any other function then Particular Integral can be obtained by resolving 1/Integral can be obtained by resolving 1/f(D)f(D) into partial fractions.into partial fractions.

ExampleExample 1: (D 1: (D22+a+a22)y=Secax. Here P.I=x/a )y=Secax. Here P.I=x/a Sinax+Cosax log(Cosax)/aSinax+Cosax log(Cosax)/a22

Lecture-12Lecture-12CAUCHY’S LINEAR EQUATIONCAUCHY’S LINEAR EQUATION

Its general form isIts general form is xxnnDDnny + …. +y = Xy + …. +y = X then to solve this equation put then to solve this equation put x = ex = ezz and and

convert into ordinary form.convert into ordinary form. ExampleExample 1: x 1: x22DD22y+xDy+y=1y+xDy+y=1 Example Example 2: x2: x33DD33y+3xy+3x22DD22y+2xDy+6y=xy+2xDy+6y=x22

Lecture-13Lecture-13LEGENDRE’S LINEAR EQUATIONLEGENDRE’S LINEAR EQUATION

Its general form isIts general form is (ax + b)(ax + b)n n DDnny +…..+y = Xy +…..+y = X then to solve this equation put ax + b = ethen to solve this equation put ax + b = ez z and and

convert into ordinary form.convert into ordinary form. ExampleExample 1: (x+1) 1: (x+1)22DD22y-3(x+1)Dy+4y=xy-3(x+1)Dy+4y=x22+x+1+x+1 Example Example 2: (2x-1)2: (2x-1)33DD33y+(2x-1)Dy-2y=xy+(2x-1)Dy-2y=x

Lecture-14Lecture-14METHOD OF VARIATION OF METHOD OF VARIATION OF

PARAMETERSPARAMETERS Its general form isIts general form is DD22y + P Dy + Q = Ry + P Dy + Q = R wherewhere P, Q, RP, Q, R are real valued functions of are real valued functions of xx.. Let C.F = Let C.F = CC11u + Cu + C22vv P.I =P.I = Au + Bv Au + Bv Example Example 1: (D1: (D22+1)y=Cosecx. Here A=-x, +1)y=Cosecx. Here A=-x,

B=log(Sinx)B=log(Sinx) Example Example 2: (D2: (D22+1)y=Cosx. Here A=Cos2x/4, +1)y=Cosx. Here A=Cos2x/4,

B=(x+Sin2x)/2B=(x+Sin2x)/2

m1 unit ii-jntuworld

Technology