BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
FIRST SEMESTER 2005 – 2006
MATH C241: MATHEMATICS III
Assignment 2 26.08.2005
2
1. Show that y1(x) = x-1 and y2(x) = x3 are solutions to x y′′ − xy′ − 3 y = 0 on the
interval (0, ∞) and give general solution
2. Determine the Wronskian of two solutions on (0, ∞) to the differential equation
xy′′ + ( x − 1) y′ + 3 y = 0
3. Solve the initial value problem
36 x′′ + 12 x′ + 37 x = 0 x(0) = 70, x′(0) = 10 .
x
4. Solve the differential equation y′′ − 4 y = 5 + e .
5. Find the particular solution of
y′′ − 4 y′ + 5 y = 0 for which y(0) = 1 and y`(0) = 5.
6. Find the particular solution of the differential equation y ′′ + 2 y ′ + y = 0, that satisfies
the initial condition y (0 ) = 2, y ′(0 ) = −1 .
7. Show that the functions y1 ( x ) = e −3 x , y 2 ( x ) = cos 2 x, and y 3 ( x ) = sin 2 x, are linearly
independent
8. Find the general solution of differential equation y ′′′ + 3 y ′′ − 54 y = 0 if one solution
of that differential equation is y = e 3 x
9. Find the particular solution of y ′′ + 4 y = 3 x 3
10. Solve the initial value problem y ′′ + 3 y ′ + 2 y = e x ; y (0 ) = 0, y ′(0 ) = 3
11. Find the particular solution of the differential equation y ′′ − 3 y ′ + 2 y = 0, that
satisfies the initial condition y (0) = 1, y ′(0) = 0 .
12. Show that the functions y1 ( x ) = x, y 2 ( x ) = x ln x, and y 3 ( x ) = x 2 , are linearly
independent on the open interval x>0
13. Find the general solution of differential equation 3 y ′′′ − 2 y ′′ + 12 y ′ − 8 y = 0 if one
2x
solution of that differential equation is y = e 3
14. Find the particular solution of y ′′ + y = sin x + x cos x
15. Solve the initial value problem y ′′ + y = cos x; y (0 ) = 1, y ′(0 ) = −1
16. If y1 = e − x be a solution of y ′′ − y = 0, fins its another solution y2, and hence write
its general solution.
17. If y1 = xe x be a solution of y ′′ − 2 y ′ + y = 0, fins its another solution y2, and hence
write its general solution.
18. Find the general solution of x 2 y ′′ − 6 xy ′ + 12 y = 0 .
19. Find the general solution of y ′′ + 4 y ′ + 5 y = 0 .
20. Find the general solution of y ′′ − y = e x , using the method of undetermined
coefficients.
21. Find the general solutions of the differential equation y ' '+3 y '−10 y = 0
22. Find the general solutions of the differential equation y ' '+6 y '+9 y = 0 .
23. Solve the initial value problem y ' '−4 y '+3 y = 0; y (0) = 7, y ' (0) = 11
24. Find general solutions of the differential equation y ' '− y '−6 y = 2 sin 3 x .
x
25. Find general solutions of the differential equation y ' '+2 y '+5 y = e sin x .
26. Find a particular solution of 2x3 y” + 3 x2 y’ + 5xy = 7
27. Find the general solution of xy” + (2x +1) y’ + ( x+ 1) y = 0
28. Solve x2 y” + 2 xy’ – 2y = 0
29. Find the general solution of y” – 6y’ + 9y = 5 e3x
30. Use, Wronskians to show that y = c1 x-2 + c2 x2 is the general solution of
x2 y” + xy’ –4y = 0 on any interval not containing zero, and find the particular
solution for which y(1) = 2 and y’(1) = 3
31. Form the differential equation for the following family of curves:
y = c1e 2 x + c 2 e −3 x
32. Find the general solution of the following equation:
y ' ' − 2 y ' + 10 y = 0
33. Find the general solution of the following equation:
y ' '−2 y ' = e x sin x
34. Show that the functions e −2 x cos 3 x and e −2 x sin 3 x are linearly independent
solutions of y ' ' + 4 y ' + 13 y = 0
35. Solve (3 − x) y ''− (9 − 4 x) y '+ (6 − 3 x) y = 0
36. Find the general solution of the differential equation (1-x2)y” – xy’ +y = 0
37. Solve y” – cot x y’ + ( cot x – 1) y = ex sin x
38.
Solve (x + 2) y” – ( 2x + 5) y’ + 2y = ( x+1) ex
39.
Solve xy” + ( x-1) y’ – y = x2
40. Solve x2 y” – 2( x2 + x) y’ + ( x2 + 2x + 2) y = 0
41.
Solve y” – 2 tan x y’+ 5y = sec x ex
2
42. Solve y” – 4x y’ + (4x2 – 3 ) y = e x
43. Solve y” +2/x y’ + a2 / x4 y=0
44. Solve y” + cot x y’ + 4y cosec 2 x = 0
45. Find by “inspection” a solution of the differential equation
xy′′ − (2 x + 1) y′ + ( x + 1) y = 0
Hence find a second LI solution of the above equation.
46. Find the solution of the Initial value problem:
y′′ − 6 y′ + 8 y = 0, y (0) = 2, y′(0) = 0
47. Find the general solution of the equation
x 2 y′′ + xy ′ − 4 y = 0
48. Find a particular solution of the equation y′′ + 9 y = sin 3 x
49. Find a particular solution of y′′ − 6 y′ + 8 y = e x
50. Find a particular solution of y′′ − 6 y′ + 8 y = x + 3