I. Riešte nasledujúce diferenciálne rovnice

1. y′ = \frac{y − 1}{x^{2}y^{2}}.  2. y′ cotgx + y = 2 y(0) = −1.  3. y′ − xy^{2} = 2xy.
4. xy′ + y = y^{2} y(0) = 0.5.  5. xy(1 + y^{2}) dx − (1 + x^{2}) dy = 0.  6. sinx dy − y lny dx = 0.
7. (x + 2y) dx − x dy = 0.  8. (y^{2} − 2xy) dx + x^{2} dy = 0.  9. y^{2} + x^{2}y′ = xyy′.
10. (x + y + 1) dx = (2x + 2y − 1) dy.  11. xy′ − y = x tg \frac{y}{x}.  12. (x^{2} + y^{2})y′ = 2xy.
13. y′ = (x + y + 2)^{2}.  14. y′ = \frac{2x − y + 1}{x − 2y + 1} .  15. y′ = \frac{y + 2}{x + y − 1} .
16. y′ = −y tgx + cos^{−1} x.  17. y′ = 2x(x^{2} + y).  18. x^{2}y′ + xy + 1 = 0.
19. xy′ + 2y + x^{5}y^{3} e^{x} = 0.  20. y′ = −2y + y^{2} e^{x} .  21. y = \frac{x}{y′} + y′.
22. y = (y′)^{2} (x + a).  23. y = (y′)^{2} x − 2 (y′)^{3} .  24. 2y′x − y = lny′.
25. y = (y′)^{2} − \sqrt{a − (y′)^{2}}, a > 0.  26. y = y′x + \frac{a}{y′} .  27. y = y′x − 3 (y′)^{3} .
28. y = y′x − 2 − y′.  29. y = y′x + a (y′)^{2} .  30. y = y′x + |y′| .
31. y′ + y^{2} − 1 = 0.  32. xy′ + 1 = e^{x−y} .  33. y = xy′ − x^{2} (y′)^{3} .
34. x^{2}y′ + xy + 1 = 0.  35. y = x(y′ − x cosx).  36. 2y′ + x = 4\sqrt{y}.
37. xy′ = y cosln \frac{y}{x}.  38. x(y′ − y) = e^{x} .  39. (y′)^{2} − 2y′ − 3 = 0.
40. y′ = e^{\frac{xy′}{y}} .  41. (y′)^{2} − 2xy′ = 8x^{2}.  42. xy′ + y = xy^{2} lnx.
43. y′ − \frac{xy}{2(x^{2} − 1)} − \frac{x}{2y} = 0, y(0) = 1.  44. xy′ − 4y = x^{2} \sqrt{y}.  45. x^{2}y′ = y^{2} + 2y + 5.
46. (3x + 5y − 7) dy = (6x + 10y + 3) dx.  47. xy′ + y = y^{2} lnx.  48. (y′)^{2} = xy.
49. x^{2} (y′)^{3} − y′ = 0.  50. y = y′x + 4y′.  51. x (y′)^{2} − yy′ + x = 0.
52. y′ = 2x(x^{2} + y).  53. xy′ − 2x^{2} \sqrt{y} = 4y.  54. (2 e^{y} −x)y′ = 1.

II. Nájdite všeobecné riešenie nasledujúcich diferenciálnych rovníc

1. y′′ = 6x − x^{−2}.  2. y′′ = \frac{1}{2y′}, y′ > 0.  3. y′′′ = \frac{2}{x}.
4. y′′ = \frac{y′}{x} + x^{2}.  5. y′′ = \frac{(y′)^{2} + 1}{x^{2} + 1} .  6. y′′ − 9y = 0.
7. y′′ + 3y′ − 4y = 0.  8. y′′ + 5y′ = 0.  9. 2y′′ − 5y′ + 2y = 0.
10. y′′ + 6y′ + 9y = 0.  11. 4y′′ + 12y′ + 9y = 0.  12. y′′ − 2a^{2}y′ + a^{4}y = 0.
13. y′′ + 16y = 0.  14. y′′ − 4y′ + 13y = 0.  15. y′′ + y′ + 2y = 0.
16. y′′ + y′ − 2y = 0.  17. 3y′′ − 2y′ − 8y = 0.  18. y′′ − 2y′ − y = 0.
19. y′′ + 6y′ + 13y = 0.  20. 4y′′ − 8y′ + 5y = 0.  21. 4y′′ − 20y′ + 25y = 0.
22. y′′′ − y′ = 0.  23. y′′′′ − 5y′′ + 4y = 0.  24. y′′′′ − 2y′′′ + y′′ = 0.
25. y′′′ − y = 0.  26. y′′′′ + 4y = 0.  27. y′′′′ + 5y′′ + 4 = 0.
28. y^{(5)} + 2y′′′ + y′ = 0.  29. y′′′′13y′′ + 36y = 0.

III. Nájdite všeobecné riešenie nasledujúcich diferenciálnych rovníc

1. y′′ − 7y′ + 10y = 40.  2. y′′ − 7y′ + 10y = 6 e^{2x} .  3. y′′ + 4y = cos2x.
4. y′′ − 7y′ + 10y = 20x^{2} − 28x + 14.  5. y′′ − 7y′ + 10y = 8 e^{2x} sinx.  6. y′′ − 4y′ + 5y = 2x^{2}.
7. y′′ − 7y′ + 10y = −(6x + 7) e^{2x} .  8. y′′ + 4y = x^{4} − 2x.  9. y′′ + 4y = cos3x.
10. y′′ − 7y′ + 10y = −12 e^{3x} .  11. y′′ + 4y = e^{−2x} .  12. y′′ + 4y = 2x sin2x.
13. y′′ + 4y = x e^{2x} sin2x.  14. y′′ − 4y′ + 5y = e^{2x} .  15. y′′ − 4y′ + 5y = sinx.
16. y′′ − 7y′ + 10y = 65 sin2x.  17. y′′ − 4y′ + 5y = e^{2x} sinx.  18. y′′ + y′ = 4x e^{x} .
19. y′′ − 4y′ + 5y = x e^{2x} cosx.  20. y′′ + y′ = x.  21. y′′ − 2y′ − 3y = e^{4x} .
22. y′′ − 4y′ + 5y = e^{2x} sin2x.  23. y′′ + y′ − 2y = 3x e^{x} .  24. y′′ − 5y′ + 4y = 4x^{2} e^{2x} .
25. y′′ − 3y′ + 2y = x cosx.  26. y′′ − 2y′ + y = 6x e^{x} .  27. y′′ + 4y′ + 4y = x e^{2x} .

28. y′′ + 4y = 5 sin3x + cos3x + sin2x.  29. y′′ − 2y′ + 2y = x^{2} + sin2x.  30. y′′ − y = 2 e^{x} −x^{2}.
31. y′′ + 3y′ − 4y = e^{−4x} +x e^{−x} .  32. y′′ + y′ − 6y = x + e^{2x} .  33. y′′ + 2y′ − 3y = x^{2} e^{x} .
34. y′′ + 2y′ + 5y = −\frac{17}{2} cos2x.  35. 2y′′ + y′ − y = 2 e^{x} .  36. y′′ + 4y′ − 5y = 1.
37. y′′′ + 3y′′ + 3y′ + y = e^{−x} sinx.  38. y′′ + 2y′ + y = e^{−x} + e^{x} .  39. y′′ + a^{2}y = e^{x} .
40. y′′′ − 3y′ + 2y = (4x^{2} + 4x − 10) e^{−x} .  41. y′′′ + y′′ = sinx + x cosx.  42. y′′′ − y = x^{3} − 1.

IV. Riešte začiatočné úlohy

1. y′′ = 2x^{3}, y(0) = 2, y′(0) = 1.  2. y′′′ = x^{−2}, y(1) = 0, y′(1) = y′′(1) = 1.
3. y′′ = \frac{y′}{x} + x e^{x}, y(0) = 2, y′(0) = 1.  4. y′′ = \frac{1}{2y′}, y(0) = 0, y′(0) = 2.
5. y′′ = \sqrt{a^{2} − (y′)^{2}}, y(0) = −1, y′(0) = 0, a > 0.  6. y′′ − 4y′ = 0, y(0) = 1, y′(0) = 3.
7. y′′ − 2y′ + y = 0, y(2) = 1, y′(2) = −2.  8. 4y′′ + y = 0, y(π) = 2, y′(π) = 3.
9. y′′′ − y′ = 0, y(0) = 3, y′(0) = −1, y′′(0) = 1.  10. y′′ + 2y′ + 5y = 0, y(0) = 2, y′(0) = 0.
11. y′′′′ + 4y = 0, y(0) = y′(0) = y′′(0) = 0, y′′′(0) = 1.  12. 4y′′ + y = 0, y(π) = 2, y′(π) = 3.
13. y′′ − 4y′ + 3y = 0, y(0) = 6, y′(0) = 10.  14. y′′ + 4y′ + 29y = 0, y(0) = 0, y′(0) = 15.
15. 4y′′ + 4y′ + y = 0, y(0) = 2, y′(0) = 0.  16. y′′ + y = sinx, y(0) = 1, y′(0) = 0.
17. y′′ + y = 4 e^{x}, y(0) = 4, y′(0) = −3.  18. y′′ − 2y′ = 2 e^{x}, y(1) = −1, y′(1) = 0.
19. 4y′′ + 16y′ + 15y = 4 e^{−3x/2}, y(0) = 3, y′(0) = −\frac{11}{2}.  20. y′′ + 2y′ + 2y = x e^{−x}, y(0) = y′(0) = 0.
21. y′′ − y′ = 2(1 − x), y(0) = y′(0) = 1.  22. y′′ − 2y′ = (x^{2} + x − 3) e^{x}, y(0) = y′(0) = 2.
23. y′′′ − 3y′ − 2y = 9 e^{2x}, y(0) = 0, y′(0) = −3, y′′(0) = 3.  24. y′′ − y = 2x, y(0) = y′(0) = 0.
25. y′′′ + 2y′′ + y′ = −2 e^{−2x}, y(0) = 2, y′(0) = y′′(0) = 1.  26. y′′ + y = sinx, y(0) = 1, y′(0) = 0.
27. y′′′ − y′ = 3(2 − x^{2}), y(0) = y′(0) = y′′(0) = 1.  28. y′′ − 5y′ + 6y = x + e^{x}, y(0) = 0, y′(0) = 1.
29. y′′′′ + y′′ = 2 cosx, y(0) = −2, y′(0) = 1, y′′(0) = 0, y′′′(0) = 0.

V. Riešte homogénne systémy lineárnych diferenciálnych rovníc

1. y′_{1} = 4y_{1} − 3y_{2}  2. y′_{1} = 7y_{1} + 6y_{2}  3. y′_{1} = y_{1} + y_{2}  4. y′_{1} = y_{1} + y_{2}
y′_{2} = 5y_{1} − 4y_{2}.  y′_{2} = 2y_{1} + 6y_{2}.  y′_{2} = 8y_{1} − y_{2}.  y′_{2} = −5y_{1} − y_{2}.
5. y′_{1} = y_{1} + 3y_{2}  6. y′_{1} = 2y_{1} + y_{2}  7. y′_{1} = y_{1} − y_{2}  8. y′_{1} = y_{1} + y_{2}
y′_{2} = −3y_{1} + y_{2}.  y′_{2} = 3y_{1} + 4y_{2}.  y′_{2} = −4y_{1} + y_{2}.  y′_{2} = −2y_{1} + 3y_{2}.
9. y′_{1} = −7y_{1} + y_{2}  10. y′_{1} = y_{1} − 3y_{2}  11. y′_{1} = 2y_{1} + y_{2}  12. y′_{1} = 3y_{1} − y_{2}
y′_{2} = −2y_{1} − 5y_{2}.  y′_{2} = 3y_{1} + y_{2}.  y′_{2} = −y_{1} + 4y_{2}.  y′_{2} = 4y_{1} − y_{2}.
13. y′_{1} = y_{1} − y_{2} + y_{3}  14. y′_{1} = y_{1} − 2y_{2} − y_{3}  15. y′_{1} = y_{1} − y_{2} − y_{3}  16. y′_{1} = 2y_{1} − y_{2} + 2y_{3}
y′_{2} = y_{1} + y_{2} − y_{3}  y′_{2} = −y_{1} + y_{2} + y_{3}  y′_{2} = y_{1} + y_{2}  y′_{2} = y_{1} + 2y_{3}
y′_{3} = 2y_{1} − y_{2}.  y′_{3} = y_{1} − y_{3}.  y′_{3} = 3y_{1} + y_{3}.  y′_{3} = −2y_{1} + y_{2} − y_{3}.
17. y′_{1} = 4y_{1} − y_{2} − y_{3}  18. y′_{1} = y_{1} − y_{2} + y_{3}  19. y′_{1} = 3y_{1} − y_{2} + y_{3}  20. y′_{1} = 2y_{1} + y_{2}
y′_{2} = y_{1} + 2y_{2} − y_{3}  y′_{2} = y_{1} + y_{2} − y_{3}  y′_{2} = y_{1} + y_{2} + y_{3}  y′_{2} = y_{1} + 3y_{2} − y_{3}
y′_{3} = y_{1} − y_{2} + 2y_{3}.  y′_{3} = −y_{2} + 2y_{3}.  y′_{3} = 4y_{1} − y_{2} + 4y_{3}.  y′_{3} = −y_{1} + 2y_{2} + 3y_{3}.

VI. Riešte začiatočné úlohy

1. y′_{1} = −5y_{1} + 2y_{2} y_{1}(0) = 1  2. y′_{1} = y_{1} − 3y_{2}  y_{1}(0) = 1  3. y′_{1} = y_{1} + 2y_{2}  y_{1}(0) = 2.
y′_{2} = −y_{1} − 7y_{2} y_{2}(0) = −1  y′_{2} = 4y_{1} − 6y_{2}  y_{2}(0) = 0  y′_{2} = 4y_{1} + 3y_{2}  y_{2}(0) = 1.
4. y′_{1} = y_{2}  y_{1}(0) = 3  5. y′_{1} = y_{2}  y_{1}(0) = 2  6. y′_{1} = −3y_{1} − y_{2} y_{1}(0) = 2.
y′_{2} = −2y_{1} + 2y_{2} y_{2}(0) = 1  y′_{2} = 12y_{1} − y_{2}  y_{2}(0) = −1  y′_{2} = y_{1} − y_{2}  y_{2}(0) = −1.
7. y′_{1} = y_{2}  y_{1}(0) = 3  8. y′_{1} = 5y_{1} + 3y_{2}  y_{1}(0) = 0  9. y′_{1} = y_{2}  y_{1}(0) = 0
y′_{2} = y_{3}  y_{2}(0) = −1  y′_{2} = −3y_{1} − y_{2}  y_{2}(0) = 2.  y′_{2} = y_{3}  y_{2}(0) = −1
y′_{3} = y_{2}  y_{3}(0) = 3.  y′_{3} = −y_{2}  y_{3}(0) = 1.

10. y′_{1} = y_{3}  y_{1}(0) = −1  11. y′_{1} = −y_{1} + y_{2} + y_{3}  y_{1}(0) = 1  12. y′_{1} = y_{2} + y_{3}  y_{1}(0) = −1
y′_{2} = −y_{1} + y_{3}  y_{2}(0) = 2  y′_{2} = y_{1} − y_{2} + y_{3}  y_{2}(0) = 0  y′_{2} = y_{1} + y_{3}  y_{2}(0) = 1
y′_{3} = −y_{2} + y_{3}  y_{3}(0) = 1.  y′_{3} = y_{1} + y_{2} + y_{3}  y_{3}(0) = 0.  y′_{3} = y_{1} + y_{2}  y_{3}(0) = 0.

VII. Nájdite štandartnú fundamentálnu maticu e^{Ax} a použite ju na riešenie počiatočných úloh y′ = Ay, y(0) = b, pričom tucnemat A a vektor b sú zadané

1. ( \begin{matrix}7 & −18 \\ 3 & −8 \end{matrix} ) , ( \begin{matrix}1 \\ −1 \end{matrix} ) .  2. ( \begin{matrix}3 & −1 \\ 4 & −1 \end{matrix} ) , ( \begin{matrix}0 \\ 2 \end{matrix} ) .  3. ( \begin{matrix}−1 & 5 \\ −1 & 1 \end{matrix} ) , ( \begin{matrix}2 \\ −1 \end{matrix} ) .

4. ( \begin{matrix}1 & 1 \\ −1 & −1 \end{matrix} ) , ( \begin{matrix}1 \\ 2 \end{matrix} ) .  5. ( \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} ) , ( \begin{matrix}1 \\ 0 \\ 1 \end{matrix} ) .  6. ( \begin{matrix}2 & 1 & −2 \\ −1 & 0 & 0 \\ 1 & 1 & −1 \end{matrix} ) , ( \begin{matrix}1 \\ 2 \\ 3 \end{matrix} ) .

VIII. Riešte nehomogénne systémy lineárnych diferenciálnych rovníc

1. y′_{1} = 3y_{1} − 2y_{2}  2. y′_{1} = −y_{1} + 5y_{2}  3. y′_{1} = y_{2} + x^{2}
y′_{2} = 2y_{1} − y_{2} + 1.  y′_{2} = −y_{1} + y_{2} + 8x.  y′_{2} = y_{1} + 2 e^{x} .
4. y′_{1} = 2y_{1} + 3y_{2} + 8 e^{x}  5. y′_{1} = −5y_{1} + 2y_{2} + e^{x}  6. y′_{1} = y_{1} − y_{2} + 2 sinx
y′_{2} = 3y_{1} + 2y_{2} + 5x.  y′_{2} = y_{1} − y_{2} + e^{2x} .  y′_{2} = 2y_{1} − y_{2}.
7. y′_{1} = 2y_{1} − y_{2}  8. y′_{1} = 2y_{1} + 4y_{2} + cosx  9. y′_{1} = −y_{1} + 2y_{2}
y′_{2} = y_{1} + 2 e^{x} .  y′_{2} = −y_{1} − 2y_{2} + sinx.  y′_{2} = −y_{1} + y_{2} + \frac{1}{cosx}.
10. y′_{1} = −y_{1} + 2y_{2}  11. y′_{1} = y_{2}  12. y′_{1} = −5y_{1} + 2y_{2} + e^{x}
y′_{2} = −2y_{1} + 3y_{2} + 15\sqrt{x} e^{x} .  y′_{2} = y_{1} + e^{x} + e^{−x} .  y′_{2} = −2y_{1} − 3y_{2} + e^{−2x} .
13. y′_{1} = y_{1} + y_{2} + 3y_{3} + 8 e^{−x}  14. y′_{1} = 2y_{1} + y_{2} − 2y_{3}
y′_{2} = −y_{1} + y_{2}  y′_{2} = −y_{1} + x
y′_{3} = −y_{1} + y_{3}.  y′_{3} = y_{1} + y_{2} − y_{3}.

IX. Riešte počiatočné úlohy

1. y′_{1} = y_{2} − 5 cosx  2. y′_{1} = 3y_{1} + 2y_{2} + 4 e^{x}  3. y′_{1} = y_{1} − y_{2} + \frac{1}{cosx}
y′_{2} = 2y_{1} + y_{2}  y′_{2} = y_{1} + 2y_{2}  y′_{2} = 2y_{1} − y_{2}
y_{1}(0) = y_{2}(0) = 0.  y_{1}(0) = 1 y_{2}(0) = −1.  y_{1}(0) = 2 y_{2}(0) = 0.