# Final Exam Math 204 (1433).

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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 ) 
sa
s2  a2
With my best wishes
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