Download Practice problems on ordinary differential equations 1. For the

c pHabala 2012
°
ODE: practice problems
Practice problems on ordinary differential equations
1. For the equation
ẋ =
x2 − x
t
solve the following Cauchy problems:
a) x(1) = 2; b) x(4) = 12 ; c) x(−1) = 32 ; d) x(1) = 43 ;
f) x(3) = 0; g) x(1) = −1; h) x(0) = 3.
e) x(−2) = 1;
Solve the following Cauchy problems:
2. x′ =
x+2
,
t
x
4. y ′ = − ,
y
6. y ′ =
1
,
y2
3. y ′ x = 1 − y,
x(2) = 4;
y(4) = −3;
5. ẋ =
e−x
,
t
y(−1) = −1;
x(1) = 0;
√
7. y ′ = 2 y ex ,
y(1) = 1;
y(0) = 4.
Find general solutions of the following equations:
8. y ′ = 6xy 2/3 ;
9. 2y ′ = 12x2 + 3x2 y 2 ;
√
10. y ′ = 4t y;
11. πxy ′ =
y′ =
12. For the equation
a) y(−1) = 3;
3y
2
−
3
x
x
sin(πy)
.
cos(πy)
solve the following Cauchy problems:
b) y(1) = 1;
c) y(0) = 3.
Solve the following Cauchy problems:
13. y ′ =
2
y + x2 sin(x),
x
15. y ′ + y + x = 0,
3
y
2
=
, y(2) = ln(3);
x+1
x−1
y
20. y ′ +
= x, y(0) = −6;
x−1
18. y ′ +
y(0) = 13;
19. ẋ = x cot(t) + 2t sin2 (t),
1
,
x
14. x′ =
y(0) = 0;
17. y ′ = 3x2 y − ex ,
21. xy ′ + y =
x
+ 1, x(0) = −2;
t+1
2xy
2x
16. y ′ + 2
= 2
, y(1) = − ln(3)
3 ;
x −4
(x − 4)2
y(π) = 2π 2 ;
¡ ¢
= 1;
x 7π
2
22. x′ = 2t3 − 2tx,
y(−1) = 0;
23. Find a general solution of the equation
24. Solve the Cauchy problem
y′ =
y′ =
x(0) = 2.
x(y − 1) √ 2
+ x + 1.
x2 + 1
y(x + 1) x + 1
−
,
x
x
y(−1) = 1 − 3e .
2
2x
y′ +
y = 1 + x2 .
2
1+x
1 + x2
a) Prove that {x, x2 − 1} is its fundamental system.
b) Find a general solution of the associated homogeneous equation.
c) Find a general solution of the given equation.
25. Consider the equation y ′′ −
1
c pHabala 2012
°
ODE: practice problems
For the following equations find their fundamental systems and general solutions:
26. y ′′ + y ′ − 6y = 0;
27. y ′′ − y ′ = 0;
28. x′′ + 9x = 0;
29. y ′′ − 3y ′ + 2y = 0;
30. x′′ − 6x′ + 9x = 0;
31. y ′′ − 2y = 0;
32. y ′′ − 6y ′ + 13y = 0;
33. ẍ + ω 2 x = 0,
34. ay ′′ − by = 0, a, b > 0;
35. x′′′ − x′′ − 2x′ = 0;
36. y ′′′ − 2y ′′ + 10y ′ = 0;
37. x′′′ − 3x′′ + 3x′ − x = 0;
40. x(4) − x′′ = 0;
41. y (4) − y = 0.
38. x′′′ − A2 x′ = 0, A > 0;
ω > 0;
39. y (4) − 2y ′′′ + y ′′ = 0;
42. For each of the following left hand-sides of linear equations with constant coefficients
a) y ′′ − 4y ′ + 3y;
b) y ′′ − 2y ′ + 5y;
c) y ′′′ − 4y ′′ + 13y ′ ;
d) y (4) + 9y ′′
Find using the guessing method a general form of a particular solution for all the following
specal right hand-sides:
α) 2ex sin(2x) + (x − 1)ex cos(2x);
β) 12 sin(3x);
γ) (x2 − 3)e2x ;
δ) (x + 1)e3x ;
ε) x2 + 1.
43. Find a general solution of
and guessing.
y ′′ + y ′ − 2y = −54x e−2x , use both variation of parameter
Solve the following Cauchy problems:
44. y ′′ − 2y ′ = ex − 5 cos(x),
45. ẍ + x = sin(t) + et sin(t),
46. y ′′ − 2y ′ + y =
et
,
t
y(0) = −1, y ′ (0) = 3;
x(0) = − 25 , ẋ(0) =
y(1) = −e, y ′ (1) = 0;
y(1) = 25 −3π +e, y ′ (1) = 1−π(2−π 2 )+e;
47. y ′′ −3y ′ +2y = 2x+(π 4 +5π 2 +4) sin(πx),
x(0) = − 81 , x′ (0) = 14 ;
48. x′′ − 2x′ = 2 sinh(2t),
49. ẍ + x =
1
,
cos3 (t)
x(π) = ẋ(π) = 12 ;
50. y ′′ − 4y = 13 sin(3x) − 5 cos(x),
51. x′′ + 4x = 9t sin(t) − 5et ,
52. y ′′ + 4y ′ + 4y =
13
10 .
−e−2x
,
x2 − 1
y(0) = 3, y ′ (0) = 1;
x(0) = −3, x′ (0) = −1;
y(0) = 1, y ′ (0) = −2.
Find general solutions of the following equations:
9
;
sin(3t)
53. y ′′ − 2y ′ = 2x − 1 + x ex ;
54. x′′ + 9x =
55. x′′ − 3x′ + 2x = sin(t) − t cos(t) − 1;
√
57. y ′′ + 2y ′ + y = 15e−x x;
56. y ′′ + 4y = 1 + 2 sin(2x);
59. ẍ + 4x = 8 sin2 (2t);
61. y ′′′ − 2y ′′ = 2ex − 1;
58. y ′′ − 2y ′ = 5 sin(x) + 10 cos(x) − 8 cos(2x);
60. x′′ − 3x′ + 2x = 2et + 2t2 − 1;
62. ẍ + 4x = −8 cot(2t).
2
c pHabala 2012
°
ODE: practice problems
Solve the following Cauchy problems:
63.
y1′ = −2y1 + 4y2
y2′ = y1 + y2
64.
y1′ = 2y1 − 3y2
y2′ = 3y1 − 4y2
65.
y1′ = y1 − 3y2
y2′ = 3y1 + y2
66.
ẋ1 = 2x1 + x2
y1 (0) = 4, y2 (0) = −1;
y1 (0) = 2, y2 (0) = 1;
y1 (0) = 1, y2 (0) = 1;
x1 (0) = 2, x2 (0) = 1;
ẋ2 = −x1 + 4x2
67.
y1′ = y1 + 4y2
y2′ = 3y1 + 2y2
y1 (0) = 3, y2 (0) = −4;
68.
ẋ1 = x1 − x2
ẋ2 = 2x1 − x2
x1 (π) = −1, x2 (π) = 0;
69.
y1′ = 2y1 + y2
y2′ = y1 + 2y2
y1 (0) = 3, y2 (0) = 1;
70.
x′1 = 3x1 − x2
x′2 = x1 + x2
x1 (0) = 1, x2 (0) = 0;
71.
y1′ = y1 − y2 + 2ex
y2′ = −y1 + y2 + ex
72.
x′1 = x1 − x2 + 9
x′2 = 10x1 − x2 + 10et
y1 (0) = 0, y2 (0) = 3;
x1 (0) = −1, x2 (0) = 6.
Bonus:
Find general solutions of the following systems:
73.
y1′ = y1 + y3
y2′ = y1 − y2
y3′ = y1 + y3
74.
y1′ = y1 + 2y3
y2′ = y1 + y3
y3′ = −y1 + y2 + 2y3
75.
x′1 = x1 − x3
x′2 = x1 + x2 + x3
x′3 = 2x1 + x2
76.
y1′ = y1 + y3 + 6ex
y2′ = y1 − y2 − 6
y3′ = y1 + y3
3