Babu Banarasi Das Northern India Institute of Technology, Lucknow

Transcription

1 Babu Banarasi Das Northern India Institute of Technology, Lucknow B.Tech Second Semester Academic Session: Question Bank Maths II (EAS -203) Unit I: Ordinary Differential Equation 1-If dy/dx + 2y tanx = Sinx & y = 0 for x = /3,show that maximum value of y is 1/8 2-Test for exactness ( e y + 1)Cosx dx + e y Sinx dy = 0 3- Solve d 2 y/dx 2 6 dy/dx + 9y = 6 e 3x +7 e -2x log 2 4- Solve d 2 y/dx 2 + dy/dx - 6y = Sin3x + Cos2x 5-Solve (D 2 + 2)y = e x Cosx + x 2 e 2x + Cosx.Coshx 6-Solve the differential equation (D 2 4D + 4)y = 8x 2 e 2x Sin2x 7- Solve d 2 y/dx 2-2 dy/dx + y = x e x sinx 8-Solve (D 2 + 2)y =tan2x 9-Solve x 2 d 3 y/dx 3 +3x d 2 y/dx 2 + dy/dx = x 2 logx 10-Solve the simultaneous equations dx/dt + dy/dt - 2y = 2 cost-7 sint and dx/dt - dy/dt +2x = 4cost - 3sint 11-Solve d 2 x/dt 2 + dy/dt +3x = e -t and d 2 y/dt 2-4 dx/dt +3y = sin2t 12-Solve by inspection method (1-x 2 ) d 2 y/dx 2 + x dy/dx y = x ((1-x 2 ) 3/ Solve d 2 y/dx 2-4x dy/dx (4x 2-1)y = -3e x2 sin2x by removing the first derivative 14-Solve d 2 y/dx 2 cotx dy/dx ysin 2 x = cosx - cos 3 x by changing the independent variable 15-Solve by method of variation of parameter d 2 y/dx 2-3 dy/dx +2y = e x /(1+e x ) 16-An inductance of 2henries and a resistance of 20 ohms connected in series with an e.m.f. 100sin 150t. If the current is zero when t = 0.find the current at the end of 0.01second. 1

2 Unit II: Series Solution 1-Solve the following equation in power series about x = 0 2x 2 d 2 y/dx 2 + xdy/dx (x+1)y = 0 2-Solve x 2 d 2 y/dx 2 + xdy/dx + (x 2-4)y = 0 3-Show that J-3/2(x) = - sinx cosx/x 4-Show that xjn = n Jn -x Jn+1 5-Prove that J3 (x)+3j0 (x) +4J0 (x) = 0 6- Prove that Pn (x) = 1/2 n n! d n /dx n (x 2-1) n 7-Prove that [P (x) ] 2 dx =2/(2n+1) 8- Prove that (n+1) Pn +1(x) = (2n+1)x Pn (x) - npn-1 (x) 9- P2n (0) = (-1) n.. ( ).. 10-Prove that Pn (x) is the coefficient of z n in the expansion of (1-2xz+z 2 ) -1/2 in the ascending powers of x. Unit III: Laplace Transform t 2, 0 < t < 2 1-Find the Laplace transform of f(t) = t-1, 2 < t < 3 7, t > 3 2-If L{f(t)} = F(s) then L{f(at)}= 1/a F(s/a). 3-Find the Laplace transform of sin ; hence find L {cos / } 4 Using Laplace transform find sin 3 t dt 5-If f(t) = (e at cosbt)/t find the Laplace of f(t) 6- Use Laplace transform to evaluate 2

3 7- Find the Laplace transform of f(t) = (1-e 2t )/t +tu(t) +cosht cost 8- Find the Laplace Transform of f(t) = t, 0 < t 2a-t, a < t 2a 9. Express the following function in terms of unit step function and hence find its Laplace transform sin,0 < < f(t)= sin2, < 10. Find the inverse laplace transform of (i) ( ) (ii) (iii) 11-Find the Inverse Laplace of (5s - 10)/(9s 2-16) 12- Evaluate L -1 [ e -s -3e -3s /s 2 ] 13-Using Convolution theorem prove that L -1 [1/s 3 (s 2 +1) ] = t 2 /2 +cost t 14- State convolution theorem for Laplace transform. Hence find the inverse Laplace Transform of ( ) 15- Solve the initial value problem using laplace transform; +9 = ( ),with initial conditions y(0) = 0, (0) =4 h ( ) = 8sin,0 < < 0, < 16- Using Laplace trans form solve the initial value problem y - 3y +3y -y = t 2 e t, y(0) = 1 y (0) = 0, y (0) = -2 Unit IV: Fourier Series & Partial Differential Equation 1- If f(x) = ( x) 2 /4,0< x <2 show that f(x) = /12+ 2-Obtain Fourier half range cosine series of the function f(t) = 2t, 0 < t < 1 2(2-t), 1 < t < 2 3-Obtain Fourier series of the function f(x) = 1 in 0 < x < and show that /8 = 1+1/ / Form partial deferential equation f(x+y+z, x 2 +y 2 +z 2 ) = 0 5-Solve P.D.E( x 3 +3xy 2 ) p+( y 3 +3yx 2 ) q = 2z ( x 2 +y 2 ) 3

4 6- Solve ( x 2 -yz) p +( y 2 -zx ) q = z 2 - xy 7- Solve r-3s +2t = e 2x+3y + sin(x+2y) + (x 2 +2xy+y 3 ) 8-Solve (D 2 DD - 2D 2 + 2D +2D) z = e 2x+3y +sin(x+2y) +(x 2 +2xy+y 3 ) 9- Solve (D 2 - D 2 ) z = (x y). 10- Solve (D- 3D - 2) 2 z = 2e 2x sin(y+3x) Unit V: Application of Partial Differential Equation 1-Clasify the following equation 2r+ 4s+3t = 0, r+4s +4t = 0 2-Solve the given P.D.E =, if u(x,0) = -x(x-1)/2 3-Solve the given P.D.E 4 + =3, =3e -x e -5x at t = 0 4-If a string of length l is initially at rest in equilibrium position and each its points is given the velocity ( )t= 0 = bsin 3 / find the displacement y(x,t) 5-.Find the temperature in a bar of length 2 whose ends kept at zero and lateral surface insulated if the initial temperature is sin x/2 + 3sin 5 x/2. 6- An insulated rod of length l has its ends A and B maintained at 0 0 Cand C respectively until steady state condition prevail. If B suddenly reduced to 0 0 C and maintained at 0 0 C, find the temperature at a distance x from A at time t. 7- Solve + = 0, 0 < x <, 0 < y <, which satisfies the conditions u(0, y ) = u(, y) = u(x, ) = 0 and u(x, 0) = sin 2 x 8- Solve the Laplace equation + = 0, in a rectangle in the xy - plane with u(x,0) = u(,b) = 0, u(0, ) and u(a, y )= f(y) parallel to y-axis. 9- Find the deflection of u(x, y, t) of the square membrane with a= b= c = 1, if the initial velocity is zero and the initial deflection f(x, y) = A sin x 2 y. 10- Solve the given partial differential equation by the method of separation of variables 2 3y = 0. 4

5 11 Solve the following equation by the method of separation of variables = where u(0, y) = 8e -3y 5