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Ordinary and Partial Differential Equations — Major (End Semester) 2021 question paper

MMMUT Electrical Engineering previous year question paper for Ordinary and Partial Differential Equations (BSM-152), semester 2, Major (End Semester) 2021. All 27 questions are listed below, each with a written answer on Nexsus.

Paper details

  • Subject: Ordinary and Partial Differential Equations (BSM-152)
  • Branch: Electrical Engineering
  • Semester: 2
  • Exam: Major (End Semester) 2021
  • Questions: 27

Questions asked in Ordinary and Partial Differential Equations Major (End Semester) 2021

  1. Q1(a). Solve the following simultaneous differential equations dx/dt - y = e^t + x, dy/dt + x = sin t, given x(0) = 0, y(0) = 0. [2 marks]
  2. Q1(b). Solve the following P.D.E. by the method of separation of variable: 2u_x + 3u_y + 5u = 0 with u(0, y) = 2e^{-y}. [2 marks]
  3. Q1(c). Solve, d^2y/dx^2 - 2x dy/dx + (x^2 + 2)y = (x^2 + 2x)e^{x^2}. [2 marks]
  4. Q1(d). State and prove Rodrigue's formula. [2 marks]
  5. Q1(e). Find the complete and singular integrals of z = px + qy + pq. [2 marks]
  6. Q1(f). Evaluate the value of I_1 and I_{-1}. [2 marks]
  7. Q1(g). Solve r + s - 2t = (y - 1)e^x. [2 marks]
  8. Q2(a). Solve d^2y/dx^2 + 3 dy/dx + 2y = e^{-x} + sin(x + 3). [5 marks]
  9. Q2(b). Solve, y" - 2y' + 2y = e^x tan x by method of variation of parameter. [5 marks]
  10. Q2(c). Solve x^2 d^2y/dx^2 + 4x dy/dx + 2y = 1/x^2 + log x sin(log x). [5 marks]
  11. Q3(a)(i). Prove that ∫_{-1}^1 (x^2 - 1)P_{n+1}P_n dx = 2n(n+1) / ((2n+1)(2n+3)).
  12. Q3(a)(ii). ∫ J_3(x)dx = -2/x J_2(x).
  13. Q3(b). Solve in series the differential equation x d^2y/dx^2 + dy/dx + y = 0. [5 marks]
  14. Q3(c)(i). Show that nP_n = xP'_n - P'_{n-1}.
  15. Q3(c)(ii). xJ'_n = nJ_n - xJ_{n+1}.
  16. Q4(a)(i). Solve (mz - ny)p + (nx - lz)q = lx - my.
  17. Q4(a)(ii). (D^2 + 6DD' + 9D'^2)z = 6x + 2y + e^{2x+y}.
  18. Q4(b). Use Charpit's method to find the complete integral of following P.D.E. (p^2 + q^2)x = pz. [5 marks]
  19. Q4(c). Find the general integral of the PDE (2xy - 1)p + (z - 2x^2)q = 2(x - yz) and the integral which passes through the lines x = 1, y = 0. [5 marks]
  20. Q5(a). A tightly stretched sting with fixed end points x = 0 and x = l is initially at rest in its equilibrium position. If it is set vibrating by giving to each of its points on initial velocity λx(l-x), find the displacement of the string at any distance 'x' from one end at any time t. [5 marks]
  21. Q5(b). Determine the solution of one-dimensional heat equation ∂u/∂t = c^2 ∂^2u/∂x^2 subject to the boundary conditions u(0,t) = 0, u(l, t) = 0, (t > 0) and the initial condition u(x, 0) = x, l being the length of the bar. [5 marks]
  22. Q5(c). An initially long uniform plate is bounded by two parallel edges and an end at right angle to them. The breadth in π, the end is maintained at a temperate 100°C at all points and other edges are at 0°C. Find the state temperature. [5 marks]
  23. QQ1. Attempt any five parts of the following.
  24. QQ2. Attempt any two parts of the following.
  25. QQ3. Attempt any two parts of the following.
  26. QQ4. Attempt any two parts of the following.
  27. QQ5. Attempt any two parts of the following.