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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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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

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posted: | 12/8/2011 |

language: | English |

pages: | 3 |

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

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