ADVANCED ENGINEERING MATHEMATICS
ADVANCEDENGINEERINGMATHEMATICS2130002 – 5th EditionDarshan Institute of Engineering and TechnologyName:Roll No.:Division :
I N D E XUNIT WISE ANALYSIS FROM GTU QUESTION PAPERS . 5LIST OF ASSIGNMENT . 6UNIT 1 – INTRODUCTION TO SOME SPECIAL FUNCTIONS . 81).METHOD – 1: EXAMPLE ON BETA FUNCTION AND GAMMA FUNCTION. 92).METHOD – 2: EXAMPLE ON BESSEL’S FUNCTION .15UNIT-2 » FOURIER SERIES AND FOURIER INTEGRAL . 163).METHOD – 1: EXAMPLE ON FOURIER SERIES IN THE INTERVAL (𝐂, 𝐂 𝟐𝐋) .184).METHOD – 2: EXAMPLE ON FOURIER SERIES IN THE INTERVAL ( 𝐋, 𝐋) .215).METHOD – 3: EXAMPLE ON HALF COSINE SERIES IN THE INTERVAL (𝟎, 𝐋) .266).METHOD – 4: EXAMPLE ON HALF SINE SERIES IN THE INTERVAL (𝟎, 𝐋) .277).METHOD – 5: EXAMPLE ON FOURIER INTERGRAL .29UNIT-3A » DIFFERENTIAL EQUATION OF FIRST ORDER . 328).METHOD – 1: EXAMPLE ON ORDER AND DEGREE OF DIFFERENTIAL EQUATION .339).METHOD – 2: EXAMPLE ON VARIABLE SEPARABLE METHOD .3510).METHOD – 3: EXAMPLE ON LEIBNITZ’S DIFFERENTIAL EQUATION .3711).METHOD – 4: EXAMPLE ON BERNOULLI’S DIFFERENTIAL EQUATION.3912).METHOD – 5: EXAMPLE ON EXACT DIFFERENTIAL EQUATION .4013).METHOD – 6: EXAMPLE ON NON-EXACT DIFFERENTIAL EQUATION.4214).METHOD – 7: EXAMPLE ON ORTHOGONAL TREJECTORY.44UNIT-3B » DIFFERENTIAL EQUATION OF HIGHER ORDER. 4615).METHOD – 1: EXAMPLE ON HOMOGENEOUS DIFFERENTIAL EQUATION .4816).METHOD – 2: EXAMPLE ON NON-HOMOGENEOUS DIFFERENTIAL EQUATION .5217).METHOD – 3: EXAMPLE ON UNDETERMINED CO-EFFICIENT.5518).METHOD – 4: EXAMPLE ON WRONSKIAN .5619).METHOD – 5: EXAMPLE ON VARIATION OF PARAMETERS .57DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
I N D E X20).METHOD – 6: EXAMPLE ON CAUCHY EULER EQUATION . 6021).METHOD – 7: EXAMPLE ON FINDING SECOND SOLUTION . 61UNIT-4 » SERIES SOLUTION OF DIFFERENTIAL EQUATION . 6222).METHOD – 1: EXAMPLE ON SINGULARITY OF DIFFERENTIAL EQUATION . 6323).METHOD – 2: EXAMPLE ON POWER SERIES METHOD. 6424).METHOD – 3: EXAMPLE ON FROBENIUS METHOD . 67UNIT-5 » LAPLACE TRANSFORM AND IT’S APPLICATION . 7025).METHOD – 1: EXAMPLE ON DEFINITION OF LAPLACE TRANSFORM . 7426).METHOD – 2: EXAMPLE ON LAPLACE TRANSFORM OF SIMPLE FUNCTIONS . 7527).METHOD – 3: EXAMPLE ON FIRST SHIFTING THEOREM . 7728).METHOD – 4: EXAMPLE ON DIFFERENTIATION OF LAPLACE TRANSFORM . 7929).METHOD – 5: EXAMPLE ON INTEGRATION OF LAPLACE TRANSFORM. 8130).METHOD – 6: EXAMPLE ON INTEGRATION OF A FUNCTION . 8331).METHOD – 7: EXAMPLE ON L. T. OF PERIODIC FUNCTIONS . 8532).METHOD – 8: EXAMPLE ON SECOND SHIFTING THEOREM . 8833).METHOD – 9: EXAMPLE ON LAPLACE INVERSE TRANSFORM . 8934).METHOD – 10: EXAMPLE ON FIRST SHIFTING THEOREM. 9135).METHOD – 11: EXAMPLE ON PARTIAL FRACTION METHOD . 9236).METHOD – 12: EXAMPLE ON SECOND SHIFTING THEOREM. 9537).METHOD – 13: EXAMPLE ON INVERSE LAPLACE TRANSFORM OF DERIVATIVES . 9638).METHOD – 14: EXAMPLE ON CONVOLUTION PRODUCT . 9739).METHOD – 15: EXAMPLE ON CONVOLUTION THEROREM . 9940).METHOD – 16: EXAMPLE ON APPLICATION OF LAPLACE TRANSFORM . 101UNIT-6 » PARTIAL DIFFERENTIAL EQUATION AND IT’S APPLICATION .10441).METHOD – 1: EXAMPLE ON FORMATION OF PARTIAL DIFFERENTIAL EQUATION 10542).METHOD – 2: EXAMPLE ON DIRECT INTEGRATION . 107DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
I N D E X43).METHOD – 3: EXAMPLE ON SOLUTION OF HIGHER ORDERED PDE . 11044).METHOD – 4: EXAMPLE ON LAGRANGE’S DIFFERENTIAL EQUATION . 11245).METHOD – 5: EXAMPLE ON NON-LINEAR PDE . 11446).METHOD – 6: EXAMPLE ON SEPARATION OF VARIABLES . 11647).METHOD – 7: EXAMPLE ON CLASSIFICATION OF 2ND ORDER PDE. 1178 GTU QUESTION PAPERS OF AEM – 2130002 . . ***SYLLABUS OF AEM – 2130002 . . . . .***DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
U ni t w i s e a na ly s i s fr om GT U q u e s ti o n p a p e r sUNIT WISE ANALYSIS FROM GTU QUESTION PAPERSUnit Number 123456W – 144282872824S – 15-3514142828W – 1533025143116S – 169282683117W – 1633021143116S – 1721538112627W – 1731335142628S – 18-2231142428Average 32527122823*GTU Weightage 4102061515*Unit weightage out of 70 marks.Unit No.Unit NameLevelGTU Hour1Introduction to Some Special FunctionEasy22Fourier Series and Fourier IntegralMedium53Differential equation and It’s ApplicationMedium114Series Solution of Differential EquationEasy35Laplace Transform and It’s ApplicationHard96Partial Differential EquationHard12DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
L i s t o f A s s i gnm e n tLIST OF ASSIGNMENTAssignment No.Unit No.Method No.4266221, 2323, 4, 543BALL METHODS53AALL METHODS65Proof of Formulae75GTU asked examples (Method No. 1 to 8)85GTU asked examples (Method No. 9 to 16)1DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
[7 ]DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
UNI T -1 » I n tro du ction To S o me Spe cia l Fu n ctio n[8 ]UNIT 1 – INTRODUCTION TO SOME SPECIAL FUNCTIONS INTRODUCTION: Special functions are particular mathematical functions which have some fixed notations dueto their importance in mathematics. In this Unit we will study various type of specialfunctions such as Gamma function, Beta function, Error function, Dirac Delta function etc.These functions are useful to solve many mathematical problems in advanced engineeringmathematics. BETA FUNCTION:1 If m 0, n 0, then Beta function is defined by the integral 0 x m 1 (1 x)n 1 dx and isdenoted by β(m, n) OR B(m, n).𝟏𝐁(𝐦, 𝐧) 𝐱𝐦 𝟏 (𝟏 𝐱)𝐧 𝟏𝐝𝐱𝟎 Properties:(1) Beta function is a symmetric function. i.e. B(m, n) B(n, m), where m 0, n 0.π(2) B(m, n) 2 02 sin2m 1 θ cos2n 1 θdθπ21p 1 q 1(3) 0 sinp θ cosq θ dθ B (2 (4) B(m, n) 0xm 1(1 x)m n2,2)dx GAMMA FUNCTION: If n 0, then Gamma function is defined by the integral 0 e xx n 1 dx and is denoted by ⌈n. ⌈𝐧 𝐞 𝐱 𝐱𝐧 𝟏 𝐝𝐱𝟎 Properties:(1) Reduction formula for Gamma Function ⌈(n 1) n⌈n ; where n 0.DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
UNI T -1 » I n tro du ction To S o me Spe cia l Fu n ctio n[9 ](2) If n is a positive integer, then ⌈(n 1) n! 12(3) Second Form of Gamma Function 0 e x x 2m 1 dx ⌈m2⌈m⌈n(4) Relation Between Beta and Gamma Function, B(m, n) ⌈(m n).π1 ⌈((5) 02 sinp θ cosq θdθ 21(2n)! π2n!4 n(6) ⌈(n ) W – 15 ; W – 16p 1q 1)⌈( 2 )2p q 2⌈( 2 )for n 0,1,2,3, 1Examples: For n 0, ⌈ π23 π2For n 1, ⌈ 2W – 1653 π24For n 2, ⌈ (7) Legendre’s duplication formula.1 π222n 1⌈n ⌈(n ) S – 161⌈(2n) OR ⌈(n 1) ⌈n (8) Euler’s formula : ⌈n ⌈(1 n) 2πsin nπ π22n⌈(2n 1);0 n 1METHOD – 1: EXAMPLE ON BETA FUNCTION AND GAMMA FUNCTIONC1Find B(4,3).𝐀𝐧𝐬𝐰𝐞𝐫:T21609 7Find B ( , ) .2 25π𝐀𝐧𝐬𝐰𝐞𝐫:2048S – 16H3State the relation between Beta and Gamma function.W – 15W – 16H4State Duplication (Legendre) formula.S – 16C57Find ⌈ .215 π𝐀𝐧𝐬𝐰𝐞𝐫:8DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002S – 16
UNI T -1 » I n tro du ction To S o me Spe cia l Fu n ctio nH6Find ⌈13.2W – 1510395 π𝐀𝐧𝐬𝐰𝐞𝐫:64T7[10 ]5 3Find ⌈ ⌈ .4 4π𝐀𝐧𝐬𝐰𝐞𝐫:2 2 ERROR FUNCTION AND COMPLEMENTARY ERROR FUNCTION: The error function of x is defined as below and is denoted by erf(x).𝐞𝐫𝐟(𝐱) 𝟏 𝛑𝐱 𝐞 𝐭 𝟐𝐝𝐭 𝐱𝟐 𝛑𝐱𝟐 𝐞 𝐭 𝐝𝐭𝟎 The complementary error function is denoted by erfc(x) and defined as𝐞𝐫𝐟𝐜 (𝐱) 𝟐 𝛑 𝟐 𝐞 𝐭 𝐝𝐭𝐱 Properties:(1) erf(0) 0(2) erfc(0) 1(3) erf( ) 1(4) erf( x) erf(x)(5) erf(x) erfc(x) 1 UNIT STEP FUNCTION (HEAVISIDE’S FUNCTION):W – 14 ; W – 16 The Unit Step Function is defined by1u(x a) {0;x a;x a. It is also denoted by H(x a) or ua(x).DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
UNI T -1 » I n tro du ction To S o me Spe cia l Fu n ctio n[11 ] PULSE OF UNIT HEIGHT:f(x) The pulse of unit height of duration T isdefined by1f( x ) {0; 0 x T.;1x Tx𝐓𝐟(𝐱) SINUSOIDAL PULSE FUNCTION: The sinusoidal pulse function is defined bysin axf( x ) {0;;πaπx a0 x 𝟏x𝛑𝐚𝟎𝐟(𝐱) RECTANGLE FUNCTION: (W – 17) A Rectangular function f(x) on ℝ is defined by11; a x b0; otherwisef( x ) {x𝐚 GATE FUNCTION:𝐛𝐟(𝐱) A Gate function fa(x) on ℝ is defined by1()fa x {0; x a; x a.1 Note that gate function is symmetric about axisof co-domain. Gate function is also a rectangle function. 𝐚𝐚DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002x
UNI T -1 » I n tro du ction To S o me Spe cia l Fu n ctio n SIGNUM FUNCTION:[12 ]f(x) The Signum function is defined by 1f( x ) ;x 00;x 0 .{ 1;x 01x 𝟏 IMPULSE FUNCTION:𝐟(𝐱) An impulse function is defined as below,0 ;f( x ) x a𝟏1; a x a εε{0 ;x a ε𝛆0 DIRAC DELTA FUNCTION(UNIT IMPULSE FUNCTION):x𝐚𝐚 𝛆W – 14 A Dirac delta Function δ(x a) is defined by δ(x a) lim f(x) .ε 0Where, f(x) is an impulse function, which is defined as0 ;f( x ) x a1; a x a ε .ε{0 ;x a ε PERIODIC FUNCTION: A function f is said to be periodic, if f(x p) f(x) for all x. If smallest positive number of set of all such p exists, then that number is called theFundamental period of f(x).DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
UNI T -1 » I n tro du ction To S o me Spe cia l Fu n ctio n[13 ] Note:(1) Constant function is periodic without Fundamental period.(2) Sine and Cosine are Periodic functions with Fundamental period 2π. SQUARE WAVE FUNCTION: A square wave function f(x) of period "2a" is defined byf( x ) {1 1;;f(x)0 x a.a x 2a1a3ax-a2a 𝟏 SAW TOOTH WAVE FUNCTION: (W – 17) A saw tooth wave function f(x) with period af(x)is defined as f(x) x ; 0 x a.axa2a TRIANGULAR WAVE FUNCTION: A Triangular wave function f(x) having period "2a" is defined byx;0 x af( x ) {.2a x ; a x 2aDARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 21300023a
UNI T -1 » I n tro du ction To S o me Spe cia l Fu n ctio n[14 ]f(x)ax-2a2aa-a3a4a FULL RECTIFIED SINE WAVE FUNCTION: A full rectified sine wave function with period "π" is defined asf(x) sin x ; 0 x π.𝐟(𝐱)𝟏 x𝟎 𝛑𝟐𝛑𝛑 HALF RECTIFIED SINE WAVE FUNCTION: A half wave rectified sinusoidal function with period "2π" is defined assin x;0 x π.f( x ) {0;π x 2π𝐟(𝐱)1 x 𝟐𝛑 𝛑𝟎𝛑𝟐𝛑𝟑𝛑𝟒𝛑DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
UNI T -1 » I n tro du ction To S o me Spe cia l Fu n ctio n[15 ] BESSEL’S FUNCTION: A Bessel’s function of 1st kind of order n is defined by xnx2x4( 1)kx n 2kJn(x) n[1 ] ( )2 ⌈(n 1)2(2n 2) 2 4(2n 2)(2n 4)k! ⌈(n k 1) 2k 0METHOD – 2: EXAMPLE ON BESSEL’S FUNCTIONC1Determine the value J1 (x).2𝐀𝐧𝐬𝐰𝐞𝐫: H2S – 162sin xπxDetermine the value J( 1) (x).2𝐀𝐧𝐬𝐰𝐞𝐫: C32cos xπxDetermine the value J3 (x).2𝐀𝐧𝐬𝐰𝐞𝐫: H42 sin x( cos x)πxxDetermine the value J( 3) (x).2𝐀𝐧𝐬𝐰𝐞𝐫: T52 cos x( sin x)πxxUsing Bessel’s function of the first kind, Prove that J0 (0) 1. DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002S – 16
UNI T -2 » Fou rie r Se rie s a nd Fou rier In te gra l[16 ]UNIT-2 » FOURIER SERIES AND FOURIER INTEGRAL BASIC FORMULAE: Leibnitz’s Formula (Take, Given polynomial function as “u”) 𝐮 𝐯 𝐝𝐱 𝐮 𝐯𝟏 𝐮′ 𝐯𝟐 𝐮′′ 𝐯𝟑 𝐮′′′ 𝐯𝟒 Where, u′ , u′′ , are successive derivatives of u and v1 , v2 , are successive integrals ofv. Choice of u and v is as per LIATE order.Where,L means Logarithmic FunctionI means Invertible FunctionA means Algebraic FunctionT means Trigonometric FunctionE means Exponential Function When Function is Exponential Function: 𝐞𝐚𝐱 𝐬𝐢𝐧 𝐛𝐱 𝐝𝐱 𝐞𝐚𝐱 𝐜𝐨𝐬 𝐛𝐱𝐞𝐚𝐱[𝐚 𝐬𝐢𝐧 𝐛𝐱 𝐛 𝐜𝐨𝐬 𝐛𝐱] 𝐜𝐚𝟐 𝐛𝟐𝐞𝐚𝐱[𝐚 𝐜𝐨𝐬 𝐛𝐱 𝐛 𝐬𝐢𝐧 𝐛𝐱] 𝐜𝐝𝐱 𝟐𝐚 𝐛𝟐 When Function is Trigonometric Function:𝟐 𝐬𝐢𝐧 𝐚 𝐜𝐨𝐬 𝐛 𝐬𝐢𝐧(𝐚 𝐛) 𝐬𝐢𝐧(𝐚 𝐛)𝟐 𝐜𝐨𝐬 𝐚 𝐬𝐢𝐧 𝐛 𝐬𝐢𝐧(𝐚 𝐛) 𝐬𝐢𝐧(𝐚 𝐛)𝟐 𝐜𝐨𝐬 𝐚 𝐜𝐨𝐬 𝐛 𝐜𝐨𝐬 (𝐚 𝐛) 𝐜𝐨𝐬 (𝐚 𝐛)𝟐 𝐬𝐢𝐧 𝐚 𝐬𝐢𝐧 𝐛 𝐜𝐨𝐬(𝐚 𝐛) 𝐜𝐨𝐬(𝐚 𝐛) NOTE (FOR EVERY, 𝐧 ℤ)π cos nπ ( 1)n sin nπ 0 cos(2n 1) 0 cos 2nπ ( 1)2n 1 sin 2nπ 0 sin(2n 1) ( 1)n2π2DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
UNI T -2 » Fou rie r Se rie s a nd Fou rier In te gra l[17 ] INTRODUCTION: We know that Taylor’s series representation of functions are valid only for those functionswhich are continuous and differentiable. But there are many discontinuous periodicfunctions of practical interest which requires to express in terms of infinite series containing“sine” and “cosine” terms. Fourier series, which is an infinite series representation in term of “sine” and “cosine” terms,is a useful tool here. Thus, Fourier series is, in certain sense, more universal than Taylor’sseries as it applies to all continuous, periodic functions and discontinuous functions. Fourier series is a very powerful method to solve ordinary and partial differential equations,particularly with periodic functions. Fourier series has many applications in various fields like Approximation Theory, DigitalSignal Processing, Heat conduction problems, Wave forms of electrical field, Vibrationanalysis, etc. Fourier series was developed by Jean Baptiste Joseph Fourier in 1822. DIRICHLET CONDITION FOR EXISTENCE OF FOURIER SERIES OF 𝐟(𝐱):(1) f(x) is bounded.(2) f(x) is single valued.(3) f(x) has finite number of maxima and minima in the interval.(4) f(x) has finite number of discontinuity in the interval. NOTE: At a point of discontinuity the sum of the series is equal to average of left and right handlimits of f(x) at the point of discontinuity, say x 0 .i. e. f(x 0 ) f(x 0 0) f(x 0 0)2DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
UNI T -2 » Fou rie r Se rie s a nd Fou rier In te gra l[18 ] FOURIER SERIES IN THE INTERVAL (𝐜, 𝐜 𝟐𝐋): The Fourier series for the function f(x) in the interval (c, c 2L) is defined by 𝐚𝟎𝐧𝛑𝐱𝐧𝛑𝐱𝐟(𝐱) [𝐚𝐧 𝐜𝐨𝐬 () 𝐛𝐧 𝐬𝐢𝐧 ()]𝟐𝐋𝐋𝐧 𝟏Where the constants a0 , an and bn are given bya0 1 c 2L f(x) dxL can 1 c 2Lnπx f(x) cos () dxL cL1 c 2Lnπxbn f(x) sin () dxL cLMETHOD – 1: EXAMPLE ON FOURIER SERIES IN THE INTERVAL (𝐂, 𝐂 𝟐𝐋)C1Find the Fourier series for f(x) x 2 in (0,2). 𝟒𝟒𝟒𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱) [ 𝟐 𝟐 𝐜𝐨𝐬(𝐧𝛑𝐱) 𝐬𝐢𝐧(𝐧𝛑𝐱) ]𝟑𝐧 𝛑𝐧𝛑𝐧 𝟏H2Find the Fourier series to represent f(x) 2x x 2 in (0,3). 𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱) [ 𝐧 𝟏C34𝐜𝐨𝐬 (𝟐𝐧𝛑𝐱𝟑𝟐𝐧𝛑𝐱) 𝐬𝐢𝐧 ()]𝟑𝐧𝛑𝟑S – 16Obtain the Fourier series for f(x) e x in the interval 0 x 2.𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱) T𝟗𝐧𝟐 𝛑𝟐(𝟏 𝐞 𝟐 )𝟐 𝐧 𝟏(𝟏 𝐞 𝟐 )𝐧𝟐 𝛑𝟐 𝟏[ 𝐜𝐨𝐬 𝐧𝛑𝐱 (𝐧𝛑) 𝐬𝐢𝐧 𝐧𝛑𝐱 ]Find the Fourier series of the periodic function f(x) π sin πx. Where 0 x 1 , p 2l 1. 𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱) 𝟐 𝐧 𝟏T5𝟒𝐜𝐨𝐬(𝟐𝐧𝛑𝐱)𝟏 𝟒𝐧𝟐Find Fourier Series for f(x) x 2 ; where 0 x 2π 𝟒𝛑𝟐𝟒𝟒𝛑𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱) [ 𝟐 𝐜𝐨𝐬 𝐧𝐱 𝐬𝐢𝐧 𝐧𝐱 ]𝟑𝐧𝐧𝐧 𝟏DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
UNI T -2 » Fou rie r Se rie s a nd Fou rier In te gra lH6[19 ] πsin 2nxShow that, π x , when 0 x π .2nn 1C7Obtain the Fourier series for f(x) (π x 22) in interval 0 x 2π.π2111Hence prove that 2 2 2 .12 123W – 14 𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱) 𝛑𝟐𝟏 𝟐 𝐜𝐨𝐬 𝐧𝐱𝟏𝟐𝐧𝐧 𝟏H8Find Fourier Series for f(x) e x where 0 x 2π. 𝟏 𝐞 𝟐𝛑𝟏 𝐞 𝟐𝛑[ 𝐜𝐨𝐬 𝐧𝐱 𝐧 𝐬𝐢𝐧 𝐧𝐱 ]𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱) 𝟐𝛑𝛑(𝐧𝟐 𝟏)𝐧 𝟏H9Find Fourier Series for f(x) eax in (0,2π); a 0 𝐞𝟐𝐚𝛑 𝟏𝐞𝟐𝐚𝛑 𝟏[𝐚 𝐜𝐨𝐬 𝐧𝐱 𝐧 𝐬𝐢𝐧 𝐧𝐱 ]𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱) 𝟐𝐚𝛑𝛑(𝐧𝟐 𝐚𝟐 )S – 18𝐧 𝟏H 10x, 0 x 1Develop f(x) in a Fourier series in the interval (0,2) if f(x) {0, 1 x 2. ( 𝟏)𝐧 𝟏( 𝟏)𝐧 𝟏𝟏()()𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟 𝐱 [𝐜𝐨𝐬 𝐧𝛑𝐱 𝐬𝐢𝐧(𝐧𝛑𝐱) ]𝟒𝐧𝟐 𝛑𝟐𝐧𝛑𝐧 𝟏C11x; 0 x 2For the function f(x) {, find its Fourier series. Hence4 x: 2 x 4111π2show that 2 2 2 .1358 𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱) 𝟏 𝐧 𝟏T12𝟒 [( 𝟏)𝐧 �Find the Fourier series for periodic function with period 2 ofπx,0 x 1f(x) {π (2 x), 1 x 2. 𝛑𝟐[( 𝟏)𝐧 𝟏]𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱) 𝐜𝐨𝐬(𝐧𝛑𝐱)𝟐𝛑𝐧𝟐𝐧 𝟏DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002W – 15
UNI T -2 » Fou rie r Se rie s a nd Fou rier In te gra lH 13Find the Fourier series of f(x) {[20 ]x2 ; 0 x π0 ; π x 2π.𝐀𝐧𝐬𝐰𝐞𝐫: 𝛑𝟐𝟐( 𝟏)𝐧 𝐜𝐨𝐬 𝐧𝐱 𝟏𝛑𝟐 ( 𝟏)𝐧 𝟐( 𝟏)𝐧 𝟐𝐟(𝐱) [ 𝟑} 𝐬𝐢𝐧 𝐧𝐱 ]{ 𝟔𝐧𝟐𝛑𝐧𝐧𝟑𝐧𝐧 𝟏CFind the Fourier Series for the function f(x) given by14 π , 0 x π1π2f( x) {. Hence show that .(2n 1) 28x π , π x 2πn 0W – 16S – 18 ( 𝟏)𝐧 𝟐𝛑(𝟏 ( 𝟏)𝐧 )()()𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟 𝐱 [𝐜𝐨𝐬 𝐧𝐱 𝐬𝐢𝐧(𝐧𝐱) ]𝟒𝐧𝟐 𝛑𝐧𝐧 𝟏C15Determine the Fourier series to representthe periodic function as shown in the figure.f(x) 𝛑𝐬𝐢𝐧 𝐧𝐱𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱) 𝟐𝐧𝝅𝐧 𝟏𝟐𝝅𝟒𝝅\pi\pix DEFINITION: Odd Function: A function is said to be Odd Function if 𝐟( 𝐱) 𝐟(𝐱). Even Function: A function is said to be Even Function if 𝐟( 𝐱) 𝐟(𝐱). NOTE:ll If f(x) is an even function defined in (– l, l), then –l f(x) dx 2 0 f(x) dx.l If f(x) is an odd function defined in (– l, l), then –l f(x) dx 0. FOURIER SERIES FOR ODD & EVEN FUNCTION: Let, f(x) be a periodic function defined in (– L, L)f(x) is even, bn 0; n 1,2,3, 𝐟(𝐱) 𝐚𝟎𝟐f(x) is odd, a0 0 an; n 1,2,3, 𝐧𝛑𝐱 𝐧 𝟏 𝐚𝐧 𝐜𝐨𝐬 (𝐋) 𝐟(𝐱) 𝐧 𝟏 𝐛𝐧 𝐬𝐢𝐧 (𝐧𝛑𝐱𝐋)DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002
UNI T -2 » Fou rie r Se rie s a nd Fou rier In te gra l Where,a0 Sr. No. Where,2 L f(x) dxL 0an [21 ]bn 2 Lnπx f(x) sin () dxL 0L2 Lnπx f(x) cos () dxL 0LType of FunctionExample x 2 , x 4 , x 6 , i. e. x n, where n is even. Any constant. e.g. 1,2, π cos ax1.Even Function Graph is symmetric about Y axis. x , x 3 , cos 𝑥 , f( x) f(x) x, x 3 , x 5 , i. e. x m , where m is odd. sin ax2.Odd Function Graph is symmetric about Origin. f( x) f(x) eax3.Neither Even nor Odd ax m bx n ; n is even & m is odd number.METHOD – 2: EXAMPLE ON FOURIER SERIES IN THE INTERVAL ( 𝐋, 𝐋)C1Find the Fourier series of the periodic function f(x) 2x.Where 1 x 1 , p 2l 2. 𝟒( 𝟏)𝐧 𝟏𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱) 𝐬𝐢𝐧(𝐧𝛑𝐱)𝐧𝛑𝐧 𝟏DARSHAN INSTITUTE OF ENGINEERING & TECHNOLOGY » » » AEM - 2130002W – 16
UNI T -2 » Fou rie r Se rie s a nd Fou rier In te gra lH2Find the Fourier expansion for function f(x) x x 3 in 1 x 1. 𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱) 𝐧 𝟏C3[22 ]𝟏𝟐( 𝟏)𝐧 𝟏𝐬𝐢𝐧 𝐧𝛑𝐱𝐧𝟑 𝛑𝟑Find the Fourier series for f(x) x 2 in l x l. 𝐥𝟐𝟒 𝐥𝟐 ( 𝟏)𝐧𝐧𝛑𝐱𝐀𝐧𝐬𝐰𝐞𝐫: 𝐟(𝐱)
Divis ADVANCED ENGINEERING MATHEMATICS 2130002 – 5th Edition Darshan Institute of Engineering and Technology Name : Roll No. : ion :
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