Download MCS 245 - Differential Equations

Çankaya University
Department of Computer Engineering
2010 - 2011 Summer Semester
MCS 245 - Differential Equations
First Midterm Examination
1) Solve the Differential Equation
y0 =
y
+ ey/x
x
2) Solve the Differential Equation
(x2 cos y + x3 cos2 y) dx − (x3 sin y + x4 sin y cos y) dy = 0
3) Solve the Differential Equation
y0 +
4
ex
y= 2 4
3x
3y x
4) Find the general solution of
y 00 + 4y =
1
sin x
5) Find the general solution of
D3 (D4 − 81)2 y = 0
where D is the differential operator.
Çankaya University
Department of Computer Engineering
2010 - 2011 Summer Semester
MCS 245 - Differential Equations
Second Midterm Examination
1) Find the solution of (1 − x2 )y 00 − 2xy 0 + 30y = 0 around x0 = 0 with initial values
y(0) = 0, y 0 (0) = 8
2) Find the general solution of x(1 − x)y 00 − 4xy 0 − y = 0 around x0 = 0.
3) a) Find the inverse Laplace Transform of F (s) =
b) Find the inverse Laplace Transform of F (s) =
s3 + 2s2 + 2
s3 (s2 + 1)
(s2
s
+ 4)2
c) Find the Laplace Transform of f (t) = t2 cos2 t
4) Solve the initial value problem
y 00 − y 0 − 2y = r(t),
where r(t) =
t
0
if
if
t<2
t>2
y(0) = 3, y 0 (0) = 4
Çankaya University
Department of Computer Engineering
2010 - 2011 Summer Semester
MCS 245 - Differential Equations
Final Examination
1) Solve the initial value problem
x2 y 00 − 3xy 0 +
15
y = 0,
4
y(1) = 2, y 0 (1) = 1
2) Find the general solution of the Differential Equation
y 00 + 4y 0 + 4y = e−2x + x
3) Find the general solution of the Differential Equation
1
(x2 + ) y 00 − xy 0 − 3y = 0
4
around x0 = 0.
4) Solve the initial value problem
y 00 − y = r(t),
where r(t) =
1
0
if
if
y(0) = 0, y 0 (0) = 0
t<3
t>3
5) Find the solution of the following system of differential equations:
dx
= 5x − 2y
dt
dy
= 4x − 4y
dt
x(0) = 2, y(0) = 4
6) Find the solution of the following system of differential equations:
dy1
5
= 2y1 − y2 − 6t − 4
dt
2
dy2
1
3
=
y1 − y2 − t − 1
dt
2
2
y1 (0) = 16, y2 (0) = 4
Çankaya University
Department of Mathematics and Computer Science
MCS 245 - Differential Equations
29.06.2011
Name-Surname:
ID Number:
CLASSWORK 1
Solve
yy 0 = (y 2 + 4) cos 3x
Çankaya University
Department of Mathematics and Computer Science
MCS 245 - Differential Equations
30.06.2011
Name-Surname:
ID Number:
CLASSWORK 2
Solve the differential equation
(3x2 ey + y 2 cos x) dx + (x3 ey + 2y sin x)dy = 0
Çankaya University
Department of Mathematics and Computer Science
MCS 245 - Differential Equations
06.07.2011
Name-Surname:
ID Number:
CLASSWORK 3
Solve the initial value problem
x2 y 00 + 11xy 0 + 25y = 0,
y(1) = 0, y 0 (1) =
3
5
Çankaya University
Department of Mathematics and Computer Science
MCS 245 - Differential Equations
07.07.2011
Name-Surname:
ID Number:
CLASSWORK 4
Solve the equation
y 00 + y 0 − 20y = 2e4x
Çankaya University
Department of Mathematics and Computer Science
MCS 245 - Differential Equations
13.07.2011
Name-Surname:
ID Number:
CLASSWORK 5
Solve the equation
y 00 + y =
1
sin x
Çankaya University
Department of Mathematics and Computer Science
MCS 245 - Differential Equations
20.07.2011
Name-Surname:
ID Number:
CLASSWORK 6
Solve the equation
y 00 + 8xy 0 + 8y = 0
around x0 = 0.
Çankaya University
Department of Mathematics and Computer Science
MCS 245 - Differential Equations
21.07.2011
Name-Surname:
ID Number:
CLASSWORK 7
Solve the equation
1
(x2 + x)y 00 + (3x + )y 0 + y = 0
2
around x0 = 0.
Çankaya University
Department of Mathematics and Computer Science
MCS 245 - Differential Equations
27.07.2011
Name-Surname:
ID Number:
CLASSWORK 8
Solve the initial value problem
y 00 + 12y 0 + 36y = t2 e−6t ,
using Laplace Transform
y(0) = 0, y 0 (0) = 4
Çankaya University
Department of Mathematics and Computer Science
MCS 245 - Differential Equations
03.08.2011
Name-Surname:
ID Number:
CLASSWORK 9
Find the solution of the following system of differential equations:
dx
= −x + 6y
dt
dy
= 2x + 3y
dt
x(0) = 1, y(0) = 4
Çankaya University
Department of Mathematics and Computer Science
MCS 245 - Differential Equations
04.08.2011
Name-Surname:
ID Number:
CLASSWORK 10
Find the solution of the following system of differential equations using matrices:
dy1
= y1 + 2y2 + 1
dt
dy2
= 4y1 − y2 + 3
dt
x(0) = 1, y(0) = 4