ADVANCED ENGINEERING MATHEMATICS

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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 ri