King Saud University MATH 225 (Differential Equations)

Department of Mathematics Final Exam

1st Semester 1436-1437 H Duration: 3 Hours

Student’s Name Student’s ID Serial Number

Question Number Mark

Question I

Question II

Question III

Question IV

Question V

Total

1

Question I:

A. Choose the correct answer.

(1) The differential equation cosx d4 ydx4

+(x2 )( dydx )6

=xy is

(a) of order 4 and nonlinear (b) of order 6 and nonlinear

(c) of order 4 and linear (d) None of the previous

(2) The undetermined coefficient method (superposition method and annihilator method) cannot be applied if f (x) in the differential equation

a d2 ydx2

+b dydx

+cy=f ( x ) , (a ,b∧care constants ) ,is equal to

(a) x2e3x (b)secx (c) ex cosx (d) None of the previous

(3) The minimum value of the radius of convergence R of a power series solution centered at

zero of the differential equation(x2−3 ) d2 ydx2

+ dydx

+xy=0 is

(a)R=√3 (b¿ R=3 (c) R=∞ (d) None of the previous

(4) The operator that annihilates (xsin2 x+ x4 e2 x) is

(a) (D¿¿2+4)(D−2)¿ (b) (D¿¿2+4)2+¿¿

(c) (D2+4 )2¿ (d) None of the previous

(5) L {tsint }=

(a) 2 ss2+1

(b)2 s

(s2+1)2 (c)

−2 s(s2+1)2

(d) None of the previous

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(6) If the differential equation d2 ydx2

+ dydx

=1 has a solution y1=1 ,then a second solution is

(a) y=x (b) y=e−x (c) y=e−2x (d) None of the previous

2

(7) L−1 { ss2−1

e−2 s}=¿

(a)cosh (t+2 )U ( t+2 ) (b) (cosht )e−2 t

(c) cosh (t−2)U (t−2) (d) None of the previous

____________________________________________________________________________________

(8) A linear differential equation with constant coefficients having solutions 5e−6x ,3 is

(a) d2 yd x2

+6 dydx

=0 (b) d2 yd x2

−6 dydx

=0

(c) d2 yd x2

+6 y=0 (d) None of the previous

________________________________________________________________________________

(9) The following conditions make the differential equation d2 ydx2

+3 dydx=0 a boundary value

problem

(a) y (3 )+2 dydx

(3 )=1 (b) y (3 )+2 dy

dx(3 )=0

2 y (4 )+ dydx(4 )=1

(c) y (3 )=1 , dydx

(3 )=2 (d) None of the previous

___________________________________________________________________________________

B. Without solving classify the differential equations below as separable, linear, exact, homogeneous and/or Bernoulli:

(i) dydx

= x y2−cosxsinxy (1−x2 )

(ii) 3 x dydx

+ (tanx ) y=0

3

(iii) ydx+x (lnx−lny )dy=0

Question II :

A. Determine the region of the xy-plane for which the differential equation has a unique solution

ex dydx

=√4− y2 .

B. Solve the initial value problem2 x2dy=(3 xy+ y2)dx , y (1 )=−2.

4

Question III:

A. Find the orthogonal trajectories of the familyx2+2xy− y2+4 x−4 y=c .

5

B. Solve the following differential equation

x2 d2 ydx2

−3 x dydx

+3 y=2 x4 sinx .

6

Question IV:

A. Solve the system of differential equations

x+ dydt

−2 y=1

dxdt −2x+4 y=e

t.

7

B. Find two linearly independent power series solutions about the ordinary point x=0,d2 ydx2

−x dydx

−2 y=0.

8

Question V:

A. Prove that if f is a piecewise continuous on ¿ and of exponential order for t>T ,thenlim ¿s→∞L { f (t ) }=0 .¿

B. Use the Laplace transform to solve the initial value problemd2 ydt 2

+2 dydt

+ y=1 , y (0 )=0 , dydt

(0 )=0.

9

Good Luck

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