Advanced Mathematics for Engineers by byrnetown70

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									Advanced Mathematics for Engineers                          Questions

  1. (a) Solve the first order linear differential equation

                                    dy
                                       + xy = x.
                                    dx
                                                                           [5]
     (b) Give a condition for the differential equation

                                  dy    M (x, y)
                                     =−          ,
                                  dx    N (x, y)

         to be exact.                                                      [2]
                                                                         k
         Find the value k such that the integrating factor u(x) = (y + x)
         makes the equation
                            dy            −y
                               =                       ,
                            dx   (y + x) ln(y + x) + y
         exact.                                                        [8]
         Hence find the solution, you may leave your answer in the form
         F (x, y) = C.
         Hint:
                                      y
                       ln(x + y) +         dy = y ln(y + x).
                                   (y + x)
                                                                       [5]




                                    1
2. (a) Show that x = 0 is a regular singular point of the differential
       equation

                        d2 y    1      dy      1
                   x2        +x   + 2x    + x−   y = 0.
                        dx2     2      dx      2
                                                                             [3]
   (b) Using the method of Frobenius we assume
                                  ∞
                            y=         an xn+r ,   a0 = 0.
                                 n=0

       Find the roots r1 , r2 of the indicial equation.                      [5]
   (c) Show that the recurrence relationship for the unknown coefficients
       is given by

                                         2n + 2r − 1
                    an = −an−1                             .
                                  (n + r − 1)(n + r + 1/2)

       Hence, find the the coefficients a1 , a2 , a3 for the two linearly inde-
       pendent solutions.                                                    [12]




                                   2
3. Consider the first order initial value problem
                          dy
                             = f (x, y), y(x0 ) = y0 .
                          dx
  The Euler method for this problem, using step size h, is given by

                          yk+1 = yk + hf (xk , yk ).

   (a) By using Taylor’s remainder theorem for y(x) about x0 and as-
       suming that y has a continuous second derivative, show that

                               |y(x1 ) − y1 | ≤ M h2 ,

       where M is to be defined.                                              [6]
   (b) For
                             f (x, y) = xy 2 , y(0) = 5,
       calculate the values yi , i = 0, 1, 2, 3, using step size h = 0.1. By
       separating the variables find the exact solution to this problem
       and hence, compute the errors in the Euler approximation.             [7]
    (c) Reduce the second order initial value problem

                   d2 y           dy               dy
                      2
                        = f (x, y, ), y(x0 ) = y0 , (x0 ) = z0 ,
                   dx             dx               dx
       into a system of two first order initial value problems. Hence,
       write down a numerical scheme for approximating the solution to
       a second order initial value problem using the Euler method.    [7]




                                   3
4. Consider the problem

                           d2 y
                       −        + 4y = 1    on (0, 1),
                           dx2
                              y(0) = y(1) = 0.

   (a) Find the exact solution to this problem.                            [6]
   (b) Given equally spaced the nodes 0 = x0 < x1 < x2 < · · · < xn = 1,
       we define
                      yi = y(xi ), i = 0, 1, 2, . . . , n − 1.
       Define the centred finite difference approximation to the second
                   d2
       derivative, dxy at the point x = xi , i = 1, 2, . . . , n − 1.
                      2

       Using n = 4 obtain the linear system of equations
                                            
             36 −16 0      y1       1
           −16 36 −16   y2  =  1               and y0 = y4 = 0.
                                 

             0 −16 36      y3       1
                                                                           [7]
   (c) Solve the linear system of equations.                               [7]




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