# quiz4

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```					BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
FIRST SEMESTER 2005 – 2006
MATH C241: MATHEMATICS III
Assignment 3                              08.09.2005

1. Using the variation of parameters method, solve the differential equation

y ′′ + y = sin x .

2. Using the variation of parameters method, solve the differential equation

y ′′ + 2 y ′ + y = e − x .

3. Using the Operator method, solve the differential equation

y ′′ − y = x 4 .

4. Obtain the normal form of 4x2 y’’ + 4xy’ + (4x2-1)y = 0
5. Using variation of parameters, find a particular solution of y’’ + y = tan x
6. Using variation of parameters, find a particular solution of
y’’ + 3 y’ + 2y = 4ex

7.   Using variation of parameters, find a particular solution of
y’’ + 9y = sin 3x
8. Using operator method find the general solution of
(D-3)2 y = e3x cos x

9. Find the complete solution of differential equation xy′′ + 3 y′ = log x , if y1=1 is
one solution of homogeneous form.
10. Find the particular solution of differential equation y ′′ − 7 y ′ + 12 y = xe − x by using
method of successive integration.
11. Find the particular solution of differential equation y ′′ − 7 y ′ + 12 y = xe − x by using
exponential shift rule
(        )
12. Obtain the normal form of given differential equation y ′′ + x 2 y ′ + 3 x 2 + 1 y = 0
13.   Find by variation of parameters a particular solution of the differential equation
y ′′ + 9 y = csc 3 x
14. Find by variation of parameters a particular solution of the differential equation
y′′ + 4 y = x sin 2 x
15.   Find by variation of parameters a particular solution of the differential equation
y′′ − 2 y′ + y = e x ln x
16.   Find by Operator methods a particular solution of the differential equation
y′′ − 3 y′ + 2 y = e2 x
17.   Find by Operator methods a particular solution of the differential equation
y′′ − 2 y′ + y = e x sin x
18.   Find by Operator methods a particular solution of the differential equation
y′′′ + y′′ + y′ + y = x 4
19.   Show that any non-trivial solution of the equation
y′′ + x 2 y = 0
has an infinite number of positive zeros.

20.   Show that any non-trivial solution of the equation
y′′ − x 2 y = 0
has at most one positive zero.

21. Find the general solution of
x2 y’’ + xy’ + y = ln x ; x> 0
22. Solve y” + y = cot x
23. Find the general solution of
xy’’ + (2+x)y’ + y = e-x
24. Find the normal form of
2x2y’’ + 2x3y’ + [(x2+1)2 – (1+x3)]y = 0;x > 0
25. Find the normal form of
4x2y’’ + 4xy’ + (4x2-9)y = 0
26. Find the normal form of
4x2y’’ + 4x3y’ + [(x2+2)2-7]y=0
27. Find the normal form of
x2y’’- 4xy’+(x4+6)y = 0; x>0
28. Find the general solution of
yiv + 4y’’’ + 6y’’ + 4y’ + y = xe-x , by operator method
29. Find the general solution of the following equation using variation of parameters
method.
y ' '−2 y ' = e x sin x

30. Find the particular solution of the following equation using operator method
y ' '−8 y '+16 y = 12e 4 x x 2

31. Find the particular solution of the following equation
( D 2 + 4) y = 4 sec 2 2 x

32. Find the general solution of the following equation
y ' ' '−2 y ' '−5 y '+6 y = e 3 x

33. Find a particular solution of

y ′′ + 9 y = 2 sec 3 x

34.   Find a particular solution of

y ′′ + y = cos ec 2 x
Questions from the text book:
Q 6(a), 6(b), 6(d) : p. 106
Q 8 : p. 135, Q 16 : p.136,         Q23 : p. 136,   Q3 : 161, Q2 : 164

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Description: r DE’s with variable coefficients Bessel functions To introduce systems of equations Systems of equations Use Laplace Transform to solve Differential Equations Laplace Transforms 32-37 To introduce Fourier Series Fourier Series 38-40 Eigenvalues and Eigen functions, Sturm Liouville Problems 40, 43 308-1 41 To introduce Partial Differential Equations Partial Differential Equations Review 42 One dim. Wave eqn 40 43 One dim. Heat eqn 41 To introduce classical methods to solve PDE’s Laplace eqn 42