Section-A This Section contain 10 questions of Multiple choice/Fill in the blanks/ True false /Maching correct answertype questions. Attempt all parts of this section. 1-Attempt all questions: (a)- The solution of the differential equation y " + y = 0 Satisfying the conditions y(0) = 1, y () = 2, is (i) y = 2 cosx + sinx (ii) y = cosx + 2 sinx (iii) y = cosx + sinx (iv) y = 2(cosx +sinx) (b)- The general solution of the equation (D4-6D3+12D2-8D) y = 0 is (i) y = c1+[c2+c3x+c4x2]e2x ( iii) y = c1+c2x+c3x2+c4x3 (c)- The particular integral of (ii) y = [c1+c2x+c3x2]e2x (iv) y = c1+c2x+c3x2+c4e2x (D2+a2) y = sinax [10 X 2 = 20 ] (i) – cos ax (ii) cos ax (iii) –cos ax (iv) cos ax (d)- when x = x0 is a singular point of the equation expressed as (i) y = d2y dy + P ( x) + Q( x) y = 0 , atleast one of its solution can be 2 dx dx ∑ an x m+n , (ii) y = ∑ an x m , (iii) y = ∑ x0x m−n , (iv) None. n =0 n=0 n =0 ∞ ∞ ∞ (e)- L { j0 (t )} is ………….. 1/(1+p2) (f)- L(cosh2t) = …… p/(p2 -4).….. −2 x (g)-The C.F. of ( D 2 − DD′ − 2 D′2 + 2 D + 2 D′) z = 0 … f1 ( y − x) + e f2 ( y + 2 x) (h)- Sin x is an even function (T/F). (i)- Cos x is an even function (T/F) (j)- Match the following:(i) = RC = RC (ii) = LC = LC (iii) = c2 (+ ) (iv) + + =0 (F) (T) Two dimensional wave equation Laplace equation Radio equations Telegraph equations Section-B (i) Solve (D2- 2. Attempt any three of the following questions. All questions carry equal marks. [10 X 3 = 30 4D+4)y =8x2e2xSin2x (ii)-If L−1 { f ( p )} = f ( t ) , then show that L−1 e − ap f ( p ) = f ( t − a ). u ( t − a ). 1 { } .... (ii)-Find the Laplace transform of e-t cos t and dt (iii)-Using Laplace transform. Its four edges x = 0. find the Laplace transform of F(t).Solve x2d2y/dx2-3xdy/dx+5y = x2Sin(logx) (ii)-Solve dx 2 + ( x − y) = 1 dt t dy 1 + ( x + 5y) = t dt t (iii)-Solve by method of variation of parameters. solve the following differential equation: +2 + 5x = e-t sin t.. t > a −1 − ap or L { e f ( p ) } = t < a. 2 1 3 5 8 d2y dy + 5x + x2 y = 0 2 dx dx (v). 0. = .. 2 ..If F (t) = . y =0 & y = b are kept at zero temperature. A thin rectangular plate whose surface is impervious heat flow has t = 0 an arbitrary (iv)-Find the solution in series of the equation x 2 distribution of temperature f (x. . about x = 0 . −π < x < 0 − x.All questions are compulsory... 0 < x < π 1 1 1 π2 + 2 + 2 + . f ( t − a ). where x(0) = 0 and x`(0) = 1. (i).Show that [ j0 ( x)] + 2[ j1 ( x)] + 2[ j2 ( x)] + 2[ j3 ( x )] + .. (i).(i) Pn ( x) = 1 dn 2 ( x − 1) n 2n n ! dx n d2y dy − 2 x + 2 y = 0 . = 1 Q5. = f ( t − a) .. u ( t − a) . Section-C Note: Attempt any two parts from each question.... f ( t − a) . ………………(1) 2 dx dx Q4. Determine the temperature at a point of the plate at t increases..... x = a .π < x < π. x2 d2y dy − 2 x(1 + x ) + 2(1 + x ) y = x 3 . [10 X 5 = 50] Q3. (iii).Find the Fourier series for f ( x) = Hence deduce that x. Q6 (i). 0.. Find the Fourier series expansion of the function f (x) = x cos x. y). (ii) Find the Fourier half range cosine series of the function .. 2 dx dx (ii)-Find the power series solution of (1 − x 2 ) 2 2 2 2 (iii). t > a Where G ( t ) = t < a. . Follow the instruction as given in each section. False/Matching correct answer type questions.cos ax (f) [xn Jn ] (i) x –n Jn+1 (x) (ii) x –n Jn-1 (x) (iii) x n Jn-1 (x) (iv) xn+1 Jn (x) Match the items on the right hand side with those on left hand side (g) (i) L [ t sin 2t] (p) (-1)n (ii) Pn ( -x) (q) eax sin bx 2 -x (iii) (D + 2D + 1) y = e cos x (r) ax (iv) e sin bx (s) y = (c1 + c2 x – cos x ) e-x (h) (i) = c2 (ii) 1 + + + +……. Find its solution under the suitable conditions 2nd paper Note: The Question paper contains three sections. Section B & Section C with the weightage of 20. Q7(i)-Using the method of separation of variables to solve = -2u (ii)-Derive one dimensional heat equation.I.cos ax (ii) cos ax (iii) . (iii) = 0 (iv) = (p) (q) z = f1(y) + x f2(y) + x3 f3(y) (r) hyperbolic (s) parabolic (iv) cos ax Indicate True or False for the following Statements: (I) (i) The Laplace’s equation in three dimension is + + = 0 True / False. (iii) Solve the partial differential equation x( y 2 + z ) p − y ( x 2 + z )q = z ( x 2 − y 2 ) where p = ∂z ∂z and q = . 3 . Attempt all parts of this section. Section A. = 2 ( 2 – t) . ∂y ∂x where u ( x .6 e4x. SECTION – A This question contains 10 Questions of multiple choice/ Fill in the blanks/ True. 0 ) = 10 e-3x. 30 & 50 marks respectively.. of (D2 + DD′ ) z = sin (x + y) is Pick the correct answer of the choices given: (e) The particular integral of (D2 + a2) y = sin ax is (i) . 0 < t < 1 1 < t < 2.(a) The number of arbitrary constants in the general solution of a differential equation is equal to the order of the differential equation. (ii) u = e-t sin x is a solution of + = 0 True / False. (b) Generating function of Pn(x) is (I – 2xz + z2)-1/2. (c) L = cot -1 (d) The P.f ( t) 2t. [10x2=20] Q1. t).1 (x) (c) Prove that [Pn(x)]2 dx = Q5.+ -………. P0(x) = x2 SECTION – B Note: Attempt any three questions.(J) (i) Laplace transform of F(t) is defined for +ve and –ve value of t.(a) If L{F(t)} = f(s). π ).e. 0 < t <a f ( t) = 2a – t.(a) A body executes damped forced vibrations given by the equation + 2k + b2 x = e-kt sin ωt Solve the equation for both the cases.(a) Find the solution in series of + x + x2 y = 0 about x = 0 (b) Prove that [xn Jn(x)] = x n Jn . All questions carry equal marks: [10x3 = 30] 3 5 Q2. (e) If a string of length L is initially at rest in equilibrium position and each of its points is given the velocity 3 t=0 = b sin . True/ False (d) Obtain Fourier series for the expansion of f(x) = x sin x in the interval ( . SECTION – C Note: Attempt any two parts from each question. Hence.π. (a) Solve x . deduce that = + .. y′ (0) = 1 Q6. (ii) Legendre’s polynomial of degree zero i. when ω 2 ≠ b2 – k2 and ω 2 = b2 – k2 (b) Solve x2 + x + y = (log x) sin (log x) (c) Apply the method of variation of parameters to solve + n2 y = sec nx Q4. find the displacement y (x. (c) Solve the differential equation using Laplace transforms method + 2 + 5y = e-x sin x where y (0) = 0. f ( t) = 2 ( 2 – t ) .erf (2√t) } = True / False. and hence prove that L {t. (ii) cosine series (b) Solve + 2 + = sin (2x + 3y) (c) Find the Fourier half range cosine series of the function 0 < t < 1 2t. [10 x 5 = 50] Q3. a < t < 2a. then prove that L{eat F(t)} = f (s-a) (b) Draw the graph and find the Laplace transform of the triangular wave function of period 2a given by t . (a) Expand for f(x) = k for 0 < x < 2 in a half range: (i) sine series.+ 4x y = x (b) Obtain the series solution of the differential equation: x2 + x + (x2 – 4) y = 0 (c) Find L {erf √t }. All questions are compulsory. 1 < t < 2 4 . Follow the instruction as given in each section. Attempt all parts of this section.a) = sin 3rd paper Note: The Question paper contains three sections. of (D2 + DD′ ) z = sin (x + y) is_______________.0) = 0 and u (x . Section A. (iii) = 0 (iv) = (p) (q) z = f1(y) + x f2(y) + x3 f3(y) (r) hyperbolic (s) parabolic (iv) cos ax Indicate True or False for the following Statements: (I) (i) The Laplace’s equation in three dimension is + + = 0 True / False.cos ax (ii) cos ax (iii) . 30 & 50 marks respectively. = 2 + u. Motion is started by displacing the string into the form y = k (Lx – x2) from which it is released at rest at time t = 0. SECTION – A This question contains 10 Questions of multiple choice/ Fill in the blanks/ True. solve where u (x. (b) Generating function of Pn(x) is _______________ (c) L _______________ (d) The P. (J) (i) Laplace transform of F(t) is defined for +ve and –ve value of t.y) = u (L. (ii) Legendre’s polynomial of degree zero i. P0(x) = x2 True/ False SECTION – B 5 .I. (a) Using the method of separation of variables. 0) = 6 e -3x . Find the displacement of any point on the string at a distance of x from one end of time t. False/Matching correct answer type questions. y) ..cos ax n (f) [x Jn Jn+1] (i) x –n Jn+1 (x) (ii) x –n Jn+1 (x) (iii) x –n Jn+1 (x) (iv) xn Jn (x) Match the items on the right hand side with those on left hand side (g) (i) L [ t sin 2t] (p) (-1)n (ii) Pn ( -x) (q) eax sin bx 2 -x (iii) (D + 2D + 1) y = e cos x (r) ax (iv) e sin bx (s) y = (c1 + c2 x – cos x ) e-x (h) (i) = c2 (ii) 1 + + + +……. (c) Using separation of variables method to solve the equation + = 0 subject to the boundary conditions u (0.. Pick the correct answer of the choices given: (e) The particular integral of (D2 + a2) y = sin ax is (i) .e. (ii) u = e-t sin x is a solution of + = 0 True / False. [10x2=20] Q1.(a) The number of arbitrary constants in the general solution of a differential equation is equal to the __________________of the differential equation. u (x. (b) A string is stretched and fastened to two points L apart. True / False. Section B & Section C with the weightage of 20.Q7. erf (2√t) } = (d) Obtain Fourier series for the expansion of f(x) = x sin x in the interval ( . π ). (b) A string is stretched and fastened to two points L apart. 0<t<1 2 (2 – t). solve = 2 + u. y′ (0) = 1 Q6.1 (x) (c) Prove that [Pn(x)]2 dx = Q5. and hence prove that L {t.y) = u (L.0) = 0 and u (x . [ 10x5 = 50] Q3.(a) Solve x . 1 < t < 2 Q7. 0<t≤ a f (t) = 2a – t. a < t < 2a (c) Solve the differential equation using Laplace transforms method + 2 + 5y = e-x sin x where y (0) = 0. Motion is started by displacing the string into the form y = k (Lx – x2) from which it is released at rest at time t = 0.y) .+ 4x3y = x5 (b) Obtain the series solution of the differential equation: x2 + x + (x2 – 4) y = 0 (c) Find L {erf √t }.a) = sin 6 . when ω 2 ≠ b2 – k2 and ω 2 = b2 – k2 (b) Solve x2 + x + y = (log x) sin (log x) (c) Apply the method of variation of parameters to solve + n2 y = sec nx Q4. where u (x. All questions carry equal marks: [10x3 = 30] Q2. u (x.+ -………. All questions are compulsory. Find the displacement of any point on the string at a distance of x from one end of time t.(a) Find the solution in series of + x + x2 y = 0 about x = 0 (b) Prove that [xn Jn(x)] = x n Jn .Note : Attempt any Three questions. SECTION – C Note : Attempt any two parts from each question. 0) = 6 e -3x . Hence.π.(a) A body executes damped forced vibrations given by the equation + 2k + b2 x = e-kt sin ωt Solve the equation for both the cases. deduce that = + . (c) Using separation of variables method to solve the equation + = 0 subject to the boundary conditions u (0. (ii) cosine series (b) Solve + 2 + = sin (2x + 3y) (c) Find the Fourier half range cosine series of the function f (t) = 2t. (a) Using the method of separation of variables. find the displacement y (x. (a) Expand for f(x) = k for 0 < x < 2 in a half range: (i) sine series. then prove that L{eatF(t)} = f (s-a) (b) Draw the graph and find the Laplace transform of the triangular wave function of period 2a given by t .(a) If L{F(t)} = f(s). (e) If a string of length L is initially at rest in equilibrium position and each of its points is given the velocity 3 t=0 = b sin .t). 7 .