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Majmaah University College of Engineering Total Marks: 40 Final Exam. Examiner: Dr. Tarek Nabil Ahmed Math. ‘204’ Differential Equations Time: 3 hour , 6/2/1433 Answer All Questions Question 1: Solve the following differential equations. (7) xP 2 - 2yP 4x 0, dx 1 1 x , dy y lny y dy (4) y y 2 tanx cos2 x, dx dy 1 (6) - y y 4lnx, dx 3x (8) (D 2 2D 4)y e x sin2x, (9) y 3y 2y 6xe x , (10) y 9y 9 tan3x (1) (siny y sinx) dx (x cosy - cosx)dy 0, (2) (3) (1 - xy) dx (xy - x 2 ) dy 0, (5) P 2 x(x - 2) P(2y - 2yx - x 2) y 2 y 0, Question 2: Find the series solution in powers of x for Airy's Equation: y xy 0 Question 3: Find the Laplace and the inverse Laplace transform of the following Functions. (1) L(e - t cos3t), 1 (4) L-1( ), 3 S 9s (2) L(t cost), 1 (5) L-1( ), 2 ( s 6) (3) L(sin 2 (t 1)) 4s 4 (6) L-1( ) 2 s 16 Question 4: Solve the differential equation by using Laplace transform. y 3 y 2 y 6e t , y(0) 3, y(0) 3 Question5 : Prove that: 1- Bernoulli equation, 2- Non Exact Differential Equations if integrating factor is a function of X, 1 s , L(coshat) 3- L(e at ) sa s2 a2 With my best wishes