PTU B.Tech CE 2nd Semester May 2019 76254 MATHEMATICS II Question Papers

Total No. of Pages : 02

Roll No.

Total No. of Questions : 09

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B.Tech. (Civil) (2018 Batch) (Sem.-2)

MATHEMATICS-II

Subject Code : BTAM-201-18

M.Code: 76254

Time: 3 Hrs.

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Max. Marks : 60

INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION - B & C have FOUR questions each.
3. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.

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4. Select atleast TWO questions from SECTION - B & C.

SECTION-A

1. Answer briefly :
   1. What is an exact differential equation? Give example.
   2. Solve p (1 + q) = qz.
   3. Classify the differential equation Uxx + Uyy=f(x, y).

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   4. Classify the singular points of x²y" +(x² – n²) = 0, n is constant.
   5. Define ordinary point of a differential equation.
   6. Write Laplace equation in spherical coordinates.
   7. Show that e^x and x are independent solutions of y'' + 2y' + y = 0 in any interval.
   8. Is xux + yuy = u² a nonlinear partial differential equation?

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   9. Write an example of linear differential equation of first order.
   10. Give an example of elliptic partial differential equation.

SECTION-B

2. a) The initial value problem governing the current i flowing in a series RL circuit when a voltage v(t) = t is applied, is given by iR+ L di/dt = t, t = 0, i (0) = 0, where R and L are constants. Find the current i (t) at any time t. (4)
   

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   b) Solve (x²D² + 7xD + 13) y = log (x) (4)
3. a) Solve by the method of variation of parameters y" – 2y' + y = e^x tan (x). (4)
   b) Obtain the series solution of the equation x² d²y/dx² +x dy/dx+(x²-4) y=0. (4)
4. a) Solve (3D² - D'²)u = sin (2x + 3y). (4)
   b) Find the complete solution of (D³ + D²D' – DD'² – D'³)z = e^x cos 2y. (4)

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5. a) Solve the partial differential equation (mz – ny) ?z/?x +(nx-lz)?z/?y=ly-mx. (4)
   b) Find the general solution of partial differential equation : (?²z/?x²) + 4 (?²z/?x?y) + 4 (?²z/?y²) = 16 log (x+2y) (4)

SECTION-C

6. a) Classify the partial differential equation (1 + y²) uxx + (1 + x²)uyy = 0 for different values of x and y. (4)
   b) Solve the equation ?u/?x +?u/?y =4, u(0,y)=8 using method of separation of variables.(4)

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7. a) Derive D'Alembert's solution of one dimensional wave equation. (4)
   b) Find the deflection of a vibrating string of unit length having fixed ends with initial velocity zero and initial deflection f(x) = a (x - x²). (4)
8. An insulated rod of length I has its end A and B maintained at 0°C and 100°C, respectively until steady state conditions prevail. If B is suddenly reduced to 0°C and maintained at 0°C, find the temperature at a distance x from A at time t. (8)
9. Solve the Laplace equation ?²u/?x² + ?²u/?y² =0 subject to the conditions u (0, y) = u (l, y) = 0, u (x, 0) = 0 and u (x, a) = sin (npx/l). (8)

NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.

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