MAT 235 Final, Spring 2005

NDU MAT 235 Ordinary Differential Equations Final Exam Duration: 2 hours Name: ________________ Section: _______________ Instructor: ____________ Grade: ________________ 1/10 1) (12 points) Solve the following differential equations: a)   0 3 2    xy dx xy x dy . b)     0 3 2 2 3     x dy xy x y dx y 2/10 2) (10 points) Solve x y y sec2 4    . 3/10 3) (8 points) Find the orthogonal trajectories to the family of curves Ce y y x 2 2 1 2     where C is an arbitrary constant. 4/10 4) (6 points) Use the convolution to find the Laplace inverse of:    1 1 2   s s G s . 5) (10 points) Solve the following initial-value problem using Laplace transform.           1 1 0 0 t e t y y t     0 0 0   y y 5/10 6/10 6) (10 points) Solve the differential system by the Laplace transform method: y x x  2   ; y x y   4  5   x 0  1 ,   2 0   y 7/10 7) (10 points) Use the elimination method to solve the linear system: t t e y x y e y x x 4 2 8 3 6         8/10 8) (14 points) Given that x0  0 is an ordinary point , find the general solution of the differential equation  0 2 2 1 2       y xy y x . 9/10 9) (20 points) Show that x0  0 is a regular singular point of the differential equation   0 1 4 8 4 2 2 2       y x x y x y , then use the Frobenius method to find the general solution. 10/10