ALLAMA IQBAL OPEN UNIVERSITY, ISLAMABAD
(Department of Mathematics)
[
WARNING
1.
2.
PLAGIARISM OR HIRING OF GHOST WRITER(S) FOR SOLVING
THE ASSIGNMENT(S) WILL DEBAR THE STUDENT FROM AWARD
OF DEGREE/CERTIFICATE, IF FOUND AT ANY STAGE.
SUBMITTING ASSIGNMENTS BORROWED OR STOLEN FROM
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“AIOU PLAGIARISM POLICY”.
Course: Ordinary Differential Equations (3496)
Level: BS (CS)
Semester: Spring, 2021
Total Marks: 100
Pass Marks: 50
ASSIGNMENT No. 1
(Units: 1–5)
Note: Attempt all questions. Each question carries equal marks.
Q.1 (a)
Verify that the indicated piecewise-defined function is a solution to the given
differential equation.
(10+10=20)
xy'2 y  0;
(b)
 x 2 , x  0
y  f ( x)   2

x  0
 x
Find the values of m so that y = xm is a solution of differential equation
x 2 y"10 y'25 y  0
Q.2 (a)
Find a solution of the given differential equation that passes through the
indicated points
x
(b)
dy
 y 2  y (i)
dx
(0,1)
(ii)
(0,0)
(iii)
1 1
( , )
2 2
Solve:
dx
 tan 2 ( x  y )
dx
Q.3 (a)
(b)
Solve the given differential equation:
(y2 cos x — 3x2y — 2x)dx + (2y sin x — x3 + ln y) dy = 0, y(0) = e
Solve:
dx
y

dx y  x'
y (5)  2
1
Q.4 (a)
(b)
Q.5 (a)
(b)
Show by computing the Wronskian that the given functions are linearly
independent on the indicated interval:
(ii) ex, -x, e4x; (–,)
(i) sin x, csc x; (0,)
Solve the given differential equation subject to the indicated conditions:
y"" — 3y"' + 3y" — y' = 0, y(0) = y' (0) = 0, y" (0) = y"' (0) = 1
Solve the differential equation using the method of Undetermined
Coefficients;
y” + y = 2x sin x
Use variation of parameters to find a particular solution of7
y" + 2y' – 8y = 2 e-2x – e-x
ASSIGNMENT No.2
(Units 5–9)
Total Marks: 100
Pass Marks: 50
Note: Attempt all questions. Each question carries equal marks.
Q.1 (a)
Find the Fourier series of f on the given interval:
–<x<
f(x)=x+,
Use the result of the above Fourier series to show that:

4
(b)
Q.2 (a)
1 1 1
   ...
3 5 7
Discuss the conditions for convergence of the Fourier series.
Determine whether the given function is even, odd or neither:
(i) f(x)=sin 3x
(ii) f(x)=x2+x
 x2
(iii) f(x)= 
2
 x
(b)
 1
 1  x  0

0  x 1 
Expand the given function in an appropriate cosine or sine series:
  ,

f(x)=  x,
 ,

Q.3 (a)
 2  x   

  x   
  x  2  
Find the interval of convergence of the given power series.

k
 k  2)
k 1
2
2
( x  4) k
(b)
Find the first four terms of a power series in x for In x sin x.
Q.4 (a)
(b)
Find solutions of 4y" + y = 0 in the form of powers series in x.
Find the charge on the capacitor in L-R-C series circuit when L = 1/2 henry,
R = 20 ohms, C = 0.01 farad, E(t) = 150 volts, q(0) = 1 coulomb, and i(0) =
0 amperes. What is the charge on the capacitor after a long time?
Q.5 (a)
Classify the given differential equation as Parabolic, Elliptic and Hyperbolic.
2
 2u
2  u

x
x 2
y 2
2
 2u
2  u
x
(iii )

x 2
y 2
(i)
(b)
y2
(ii ) x 2
2
 2u
 2u
2  u


xy
y
2
x 2
xy
y 2
Show that Laplace's equation in three dimensions,
satisfied by the function:
1
u  1 /[( x  a) 2  ( y  b) 2  ( z  c) 2 ] 2
3
 2u  2u  2u
 0 is


x 2 y 2 z 2