MAT 235 Final, Fall 2002

NDU MAT 235 Ordinary Differential Equations Final Exam Duration: 55 minutes Name: ________________ Section: _______________ Instructor: ____________ Grade: ________________ 1/14 1) (12 points) Solve the following differential equations. a)     0 2 2    xy dy y dx x 2/14 b) 4 2 xy x y y    3/14 2) (8 points) Find the family of orthogonal trajectories of the family of curves x c y ln  , for x  0 . 4/14 3) (5 points) Given        t v v dv t ve t f 0 sin , find the Laplace transform    L f t  . 5/14 4) (9 points) Find the general solution of the following system of differential equations. 2 1 y  y  t y y     1 2 4 6/14 5) (12 points) Solve the following differential equation: 1 2 4 4 3       x y xy x y , x  0 . 7/14 6) (6 points) Given the following clairaut’s differential equation: y y y xy y        ln . a) Find a general solution. b) Find a singular solution. 8/14 7) (6 points) Solve the following differential equation   0 4 4 4     y y y . 9/14 8) (12 points) Use Laplace transform to find the solution of the system 0 3 2 1 3 4 2 1 2 2 1 1         y y y y y y with     0 0 0 2 1   y y 10/14 (Use this Page to continue problem 8) 11/14 9) (12 points) Solve the following initial-value problem using Laplace transform.           1 56 1 0 7 6 t t y y y with     0 0 0 0    y y 12/14 (Use this Page to continue problem 9) 13/14 10) (18 points) Consider the differential equation     0 1 3 2        y y x x y x . a) Show that x0  0 is a regular singular point. b) Use Frobenius theorem to find two linearly independent solutions   y1 x and   y2 x . 14/14 c) Find a general solution for the given differential equation.