VIEWS: 75 PAGES: 4 CATEGORY: Education POSTED ON: 1/9/2010
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] 4