MTH Feburary 2012 Final term PAPER SOLVED TODAY s Paper

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1 MTH401 7 Feburary 01 Final erm PAPER SOLVED TODAY s Paper Toal Quesion: 5 Mcqz: 40 Subjecive quesion: 1 4 q of 5 marks 4 q of 3 marks 4 q of marks Guidelines: Prepare his file as I included all pas papers and curren papers (shared ill now) in i. You will have o clear he conceps and formulas of opics according o which quesons are solved in file. Objecive: MCQz Topic TODAY s PAPER no 1 Number of Mcqz Raio Tes Convergence 5 Divergence D.E(Inegraing Facors 7 +Homogenous+linear+bernoli) Z= X + Z 1 Reacance & Impedence 1 Damped Moion Maxima 1 Quasi period 3 Besslen s Equaion 1 Marix Type(square+sysem o marix conversion) Eigen Values+Eigen Vecor 4 Muliplicy of Eigen Vecor 3 6

2 D.E operaor General Soluion 1 BVP 1 Please review he formulas of above opics. Q:1 dx dy - 5x + = 5 e d d dx dy - x + = e d d lec 36 example 1 in decoupled form. dx dy - 5x + = 5e d d dx dy - x + = e d d ( ) ( 1) D - 5 x + Dy = 5e D - x + Dy = e D - 5 D 5e D D - 5 5e Deerminans are,, D -1 D e D D -1 e Therefore, in decpoupled form, we ge D - 5 D 5e D x D -1 D e D D - 5 D D - 5 5e y D -1 D D -1 e Q:

3 Find order of homogenous equaion obained from non homogenous differenial equaion: y y y x = 4 + 5?? ( MARKS) Find he eigenvalues of he following sysem Ê3-9ˆ X Á X Ë4-3 Soluion: X ' Ê 3-9ˆ Á X 4-3 Ë Ê 3-9ˆ A Á Ë 4-3 for eigen values, A - li = l l ( - l )(- - l) + = ( l ) l ( l ) 9 3l 3l l l = = = l + 7 = 0 l = 7 i and - 7 i are he wo complex eigen values Q:3 Wha is Chemical reacion firs order equaion? () Page no 100 Answer: Q:4 Wha is characherisic equaion? Page no 379

4 Answer: Q:5 Can we exend power series? AnsweR: Page no 68 I answered in yes and hen wroe he exended form of power series. Q:6 Page no 371 Q:7 Wrie sysem of equaion in marix form? Soluion: Page no 387

5 dx = - 3x + 4y - 9z d dy d dz = 6x - y = 10x + 4y + 3z d Soluion : Èdx Í d Í È Èx Ídy = Í Í y Í d Í Í Í Í Íz dz Î Î Í ÍÎ d Q:8 Page no 98 Dudce special case of logisic equaion (epidemic spread)? (5) Q:9 Find order of homogenous equaion obained from non homogenous differenial equaion: y y y x = 4 + 5?? ( MARKS)

6 Q:10: Find a series soluion for he differenial equaion such ha y + y = 0 abou Find condiion of cofficen foran & an ( cn & cn ) + +? Q:11 Which series is idenically zero? Page no 73 Answer: Q:1 È3-18 A Í 9 Î - eigen values? Eigen vecors? Noe: I am no going o solve his quesion solve i by your self by consuling wo examples below. ========================Firs Paper End===================== Q1: Find Coefficien of merix: dx 3x y d = - -

7 dy 5x 7y d = + Soluion: Cofficen of marix = A È-3 - Í 5 7 Î Q: Eigen Values of merics. È A= 3 Í 0 3 Î Consider he quesion below: Ê 3-9ˆ A Á 4 3 Ë - for eigen values, A - li = 0 3- l l ( - l)( - - l) + = ( l) l ( l ) = = 9 3l 3l l = 9 l 36 0 l + 7 = 0 l = 7 i and - 7 i are he wo complex eigen values This quesion is similar o above. Q3: wheher or no a singular poins have real number if no hen give some examples?

8 Answer: Page no 84 Q4: Solve he differenial equaion. 1 dy 1 y dx Soluion: 1 dy y dx dy y Ú dy y 1 ( 1) Ú ( 1) ln y = x + c x c y e + dx dx Q5: complemenary soluion of DE '' ' y - 4y + 4y = e x Soluion: Page no 18 Q6: sae he Bessel s funcion of firs kind of order ½ and -1/. Soluion: Page no 313

9 Only pu he value of ½ in J v (x) and - ½ in J -v (x) a he places of v. Q7: Define he derivaive of A () = Èe Í Í Í8 Î Answer: Repeaed Q8: Find he egien values of A È Í1-1 Í Í Í 4 1 Í - ÍÎ 9 3 Soluion: Consider he quesion below.

10 Ê3-9ˆ A Á Ë4-3 for eigen values, A - li = 0 3- l l ( - l )(- - l) + = ( l) l ( l ) = = 9 3l 3l l = 9 l 36 0 l + 7 = 0 l = 7 i and - 7 i are he wo complex eigen values Q9: bh lamba ha mery sy noe ni hoa ime hora ha is lia L Q10: Find he auxiliary soluion of x = 3x - y -1 and = y y x e Consul page no 141 Q11: Wrie down he sysem of differenial equaions (5marks) dx dy = 6x + y + 6, = 4x + 3y d d In form of X ' = AX + F( ) Soluion: X ' = Ê Á ˆ X + Ê Á ˆ Ë4 3 Ë

11 ====================== PAST PAPERS=========================== Q: An elecronic componen of an elecronic circui ha has he abiliy o sore charge and opposes any change of volage in he circui is called Inducor Resisor Capacior None of hem Q: If Ao is iniial value and T denoes he half-life of he radioacive subsance han T da d 1 A KA A ( ) 0 A T None of he above dy dx + + is Q: inegraing facor of he given equaion xcos x y( xsin x cos x) Xsecx Cosx Cox Xsinx Q: Operaor mehod is he mehod of he soluion of a sysem of linear homogeneous or linear non-homogeneous differenial equaions which is based on he process of sysemaic eliminaion of he Dependen variables

12 Independen variable Choice variable None of hem Q: If E () =0, R =0 Elecric vibraion of he circui is called Free damped oscillaion Un- damped oscillaion Over damped oscillaion None of he given Q: Eigen value of a marix Ê 3 4ˆ Á Ë , 5 10, 5 5, 5 None Ê Q: Eigen value of a marix 1 1 ˆ Á Ë1 1,0 1,1 1, None Q: For Eigen values l 5,5 of a marix A Ê 3 4 Á ˆ Ë-1 7,here exiss... Eigen vecors. infinie one wo hree

13 Q: If a marix has 1 row and 3columns hen he given marix is called Column marix Row marix Recangular marix None Q: The general soluion of differenial equaion e y x y x e e e x y x - y cx cy cx cx dy x + y dx x.is given by dy x Q: The inegraing facor of he D.E y ln y ye is dx + = x e e y e 1 x e x y Q: For he equaion of free damped moion dx d 1 1 dx + + = he roos are d l w x 0 m = - l + l + w & m = -l - l + w if l -w >0 Then he equaions said o be:

14 Under damped Over damped Criically damped None of hem dy dx Q: For he sysem of differenial equaions = x, = 3y he independen variable is d d (Are) X, Y, X,y dy dx Q: For he sysem of differenial equaions = x, = 3y he dependen variable is d d (Are) X, Y, X,y Ê 4 - l 1 0 ˆ Á Q: 0 4 l 1 - = 0 gives Á l Ë l l l 4 of mulipliciy of 1 4 of mulipliciy of 4 of mulipliciy of 3

15 None of he given. Q: wronksin of x, x is x X O None of he above a) Marix A nd value of lembda was given o find he eigen vecor? 3 marks. Answer: (This quesion is solved by Shining Sar as original quesion was missing so I pu i here for reference.) Ê -3 1 ˆ Á Ë -4 A=, corresponding Eigen valuel = -. Ê -3 - (-) 1 0ˆ Á Ë -4 - (-) 0 Ê ˆ Á Ë - 0 Add wo imes row 1 in row Ê ˆ Á Ë k + k = 0 k 1 k 1 Choosing k = 1, we ge k = 1 1 Ê1ˆ herefore, eigen vecor is V Á Ë1 b) X =AX was given o find he eigenvalue and Eigen vecor? 5 marks. (This quesion is solved by Shining Sar as original quesion was missing so I pu i here for reference.)

16 For eigen values consu his quesion and for eigen vecor look a he above. Ê3-9ˆ X Á X Ë4-3 Soluion: Ê 3-9ˆ X ' Á 4 3 X Ë - Ê 3-9ˆ A Á Ë 4-3 for eigen values, A - li = l l ( - l )(- - l) + = ( l ) l ( l ) = = 9 3l 3l l = 9 l 36 0 l + 7 = 0 l = 7 i and - 7 i are he wo complex eigen values c) Solve DE dy-7dx=0 for iniial value f(0)=1? 5 marks. Answer:

17 dy - 7dx = 0 dy 7dx Ú dy Ú 7dx y = 7( x ) + c f (0) 1 f (0) = 7(0) + c f (0) = 0 + c 1 C y = 7x + 1 d) Find he general soluion of 4x^ y '' + 4xy' (4x^-5)y=0 (i is he Bessel's Equaion a nd same quesion is given in exercise pg 314 of our handous)? 5 marks Answer:

18 Answer: e) When a funcion is said o be analyic a any poin? marks Answer: A funcion is said o be analyic a poin if he funcion can be represened by power series in (x a) wih a posiive radius of convergence. f) Wha is he raio es? (is on pg 64 of our handous) 5 marks g) Wha is he formula for radius of convergence? (Is on pg 65 of our hndous) marks Answer: h) Wrie sysem of linear differenial equaions for wo variables x and y? (is on pg 333 of our handous). marks i) wrie any 3 D.Es of order? 3 marks Page no 07 Answer:

19 j) D.E was given o conver in normal form? 3 marks Answer: k) Any example of boundary value problem? marks

20 Noe: Power series sy ziada NHI ha. Lecure 35 o 45 pr ziada emphsis ha Q No marks: Wrie annihilaor operaor for x+3xe^ (6x) e ki power 6 xs Q No marks: Wrie he soluion of simple harmonic moion in alernaive simpler form x()=c1cosw+csinw from lec page 199

21 Answer: Q No marks: Define general linear DE of nh order Answer: Define elemenary row operaion. Answer: Addiion or muliplicaion of wo rows. Eigenvalue of mulipliciy m 3 Answer:

22 Fundamenal of marix 3 Answer: Wha is deerminnan? How o find i. Wrie equaion in marix form.

23 Find general soluion... 5marks.. Forbenius Theorem... 5marks Super posiion mehod for vecors Answer: Explain convergence and infiny condiion of a infinye sereies. Wha does hese symbols mean? Q. Solve he sysem of differenial equaions eliminaion. dy dx = x, = y d d by sysemaic

24 Soluion: dy = x Dy - x = 0 d...( i) dx = y - y + Dx = 0 d...( ii) Operae (ii) by D, we ge - Dy + D x = 0...( iii) Add (i) and (iii), we ge Dy - x = 0 - Dy + D x = D x ( D ) Auxiliary equaion is m - 1 = 0 ( ) x ( ) - x = 0-1 x = 0 c e 0 1 Pu his in (i), we ge - Dy - È Îc1e + ce = c e Dy = c e + c e m = ± 1 Inegrae boh sides, we ge y = c e - c e - y + y = 0 Q3. Find a series soluion for he differenial equaion abou such ha and n y( x) Â a nx n 0 Soluion:

25 an an+ = - ; n = 0,1,,... ( ) ( n + )( n + 1) a0 a0 For n = 0, a = - = ( + )( + ) a1 a1 For n = 1, a3 = - = For n, ( + )( + ) a a 1 Ê a ˆ a Á Ë = a4 = - = - = - - = ( )( ) a3 a3 1 Ê a1 ˆ a1 For n = 3, a5 = - = - = - Á - = Ë 6 10 ( )( ) y x = a + a x + a x + a x + a x + a x = Ê a ˆ Ê a ˆ Ê a ˆ Ê a ˆ y ( x) a a x Á x Á x 6 Á 4 x Á x Ë Ë Ë Ë10 Ê ˆ Ê ˆ y ( x) = a0 Á1 - x + x a1 Á x - x + x +... Ë 4 Ë Q4. Wrie soluion 4 5 X = Cos3 - Sin3 3 3 X = A. Sin( w + f) in he form. Ê 4 ˆ Ê 5 ˆ 41 A = Á + Á - = Ë 3 Ë 3 3 f Ê 4 / 3 ˆ Á Ë -5 / 3-1 = an = radians 41 sin ( ) = ( + ) x Q5. Case 1:

26 is a funcion of y Case: Then find he inegraing facor in boh cases. Soluion: u 1 xm + yn Q8. Under which condiions linear independence of he soluions for he differenial equaion y + P( x) y + Q( x) y = 0...(1) is guaraneed? Soluion: Linear independence is guaraneed in case when he Wronskian of he wo soluions is no equal o zero. Q10. When Frobenius Theorem is used in Differenial a ( x) y + a ( x) y + a ( x) y = 0 Equaion 1 0? Soluion: When we have a regular singular poin x= x0, hen we can find a leas one series n+ r soluion of he form y = Â cn ( x - x0 ) n 0 deermine afer solving he differenial equaion., where r is he consan ha we will Q1. Define Legendre s polynomial of degree n Soluion:

27 Legendre polynomial is an n h degree polynomial and i is given by he formula n 1 d Pn x x n n n! dx ( ) = ( -1) n Q13. Wha is he ordinary differenial equaion and give an example? Soluion: A differenial equaion which only includes ordinary derivaives is known as ordinary differenial equaion. Some examples of ordinary differenial equaions include: dy dx dy d y = x + y ( x 1)( y 1) = + + dx dy + + 3y = 0 dx dx