BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE PILANI Work Integrated Learning Programmes Division B.S. Process Engineering Second Semester 2012-2013 Assignment I PEAB ZC131 Marks: 200 Engineering Mathematics – I Weightage : 20% 1. Find vector equation and parametric equations for the line passing through the points P1(1, 2, 1) and P2(3, 5,-2). (Points are given in terms of their x, y, z components.) [8 marks] 2. Find the equation of a plane containing the point (5, 1, 3) and perpendicular to the vector 2ˆi-3ˆj+4ˆ k. [8 marks] 3. Find the equation of a plane containing three points (1, 0, 2), (3, 1, 4), (-1, 2, 0). [10 marks] 4. Solve the following first order linear differential equations: (a) dy dx = 2 sin 2x (b) dy dx = xe-x (c) dy dx = y [4 + 5 + 5 = 14 marks] 5. Solve the differential equations below. (a) dy dx = 0.2×2 + y (b) y3 dy dx + x3 = 0 (c) xdy dx = x + y [5 + 5 + 8 = 18 marks] 6. Solve the following exact differential equations to obtain y as a function of x. (a) x3dx + y3dy = 0 (b) sin x cos y dx + cos x sin y dy = 0 (c) ey (sinh x dx + cosh x dy) = 0 [24 marks] 7. If a, b, p, q are constants, under what condition does the equation (ax+by)dx+(px+qy)dy = 0 becomes an exact differential equation. Solve this exact differential equation. [10 marks] 8. Solve the following differential equations by making them exact differential equations. (a) 3x2y dx + 4x3dy = 0 (b) (ex+y + yey)dx + (xey – 1)dy = 0 [20 marks] 9. Solve the non-homogeneous differential equations below. (a) dy dx + y sin x = ecos x (b) dy dx = (y – 2) cot x [20 marks] 10. Solve the initial value problem dy dx + 4xy = e-2x with y(0) = 1. [10 marks] 11. Solve the initial value problem dy dx + 36y = 0 with y(0) = 0 and dy dx|x=0 = 5. [10 marks] 12. If sin x, cos x, x sin x and x cos x are four possible solutions to a fourth order linear differential equation calculate the Wronskian and check whether these four solutions are independent. [8 marks] 13. Solve the following differential equations. (a) d2y…