Copyright © 2010 Astashova I.V., Nikishkin V.A.
Practicum on course "Differential equations". Tutorial. 3rd edition, revised. Moscow: Publishing Center of EOI, 2010. 94 p., illustrated. URL:
Task 22 got additional parentheses for ln(). The equilibrium points are not convenient otherwise.
| 1. \( \left\{ \begin{aligned}\dot{x} &= \left(x + y\right)^{2} - 1 \\\dot{y} &= - x - y^{2} + 1 \end{aligned}\right. \) |
11. \( \left\{ \begin{aligned}\dot{x} &= \sqrt{x^{2} - y + 2} - 2 \\\dot{y} &= \operatorname{arctg}{\left (x \left(x + y\right) \right )} \end{aligned}\right. \) |
21. \( \left\{ \begin{aligned}\dot{x} &= x y - 4 \\\dot{y} &= - \left(x - 4\right) \left(x - y\right) \end{aligned}\right. \) |
| 2. \( \left\{ \begin{aligned}\dot{x} &= - x^{3} - 2 y \\\dot{y} &= 3 x - 4 y^{3} \end{aligned}\right. \) |
12. \( \left\{ \begin{aligned}\dot{x} &= \sqrt{\left(x - y\right)^{2} + 3} - 2 \\\dot{y} &= - e + e^{- x + y^{2}} \end{aligned}\right. \) |
22. \( \left\{ \begin{aligned}\dot{x} &= x^{2} - y \\\dot{y} &= \log{\left (x^{2} - x + 1 \right )} - \log{\left (3 \right )} \end{aligned}\right. \) |
| 3. \( \left\{ \begin{aligned}\dot{x} &= x + y + 1 \\\dot{y} &= y + \sqrt{2 x^{2} + 1} \end{aligned}\right. \) |
13. \( \left\{ \begin{aligned}\dot{x} &= - \sqrt{- x^{2} + y + 4} + 3 \\\dot{y} &= \log{\left (x^{2} - 3 \right )} \end{aligned}\right. \) |
23. \( \left\{ \begin{aligned}\dot{x} &= y \\\dot{y} &= \sin{\left (x + y \right )} \end{aligned}\right. \) |
| 4. \( \left\{ \begin{aligned}\dot{x} &= x^{2} - y \\\dot{y} &= \left(x - y\right) \left(x - y + 2\right) \end{aligned}\right. \) |
14. \( \left\{ \begin{aligned}\dot{x} &= 2 x + y^{2} - 1 \\\dot{y} &= 6 x - y^{2} + 1 \end{aligned}\right. \) |
24. \( \left\{ \begin{aligned}\dot{x} &= \log{\left (- x + y^{2} \right )} \\\dot{y} &= x - y - 1 \end{aligned}\right. \) |
| 5. \( \left\{ \begin{aligned}\dot{x} &= \log{\left (\frac{y^{2}}{3} - \frac{y}{3} + \frac{1}{3} \right )} \\\dot{y} &= x^{2} - y^{2} \end{aligned}\right. \) |
15. \( \left\{ \begin{aligned}\dot{x} &= \left(2 x - y\right)^{2} - 9 \\\dot{y} &= - \left(x - 2 y\right)^{2} + 9 \end{aligned}\right. \) |
25. \( \left\{ \begin{aligned}\dot{x} &= \left(x + 3\right) \left(x - y\right) \\\dot{y} &= y - 1 \end{aligned}\right. \) |
| 6. \( \left\{ \begin{aligned}\dot{x} &= 2 x y - 4 y - 8 \\\dot{y} &= - x^{2} + 4 y^{2} \end{aligned}\right. \) |
16. \( \left\{ \begin{aligned}\dot{x} &= - e^{x} + e^{y} \\\dot{y} &= \sqrt{3 x + y^{2}} - 2 \end{aligned}\right. \) |
26. \( \left\{ \begin{aligned}\dot{x} &= y^{2} - 1 \\\dot{y} &= \left(x - 1\right) \left(x - 2 y\right) \end{aligned}\right. \) |
| 7. \( \left\{ \begin{aligned}\dot{x} &= - 4 x - 2 y + 4 \\\dot{y} &= x y \end{aligned}\right. \) |
17. \( \left\{ \begin{aligned}\dot{x} &= 2 x + y^{2} - 1 \\\dot{y} &= 6 x - y^{2} + 1 \end{aligned}\right. \) |
27. \( \left\{ \begin{aligned}\dot{x} &= - \left(x - y\right) \left(y - 1\right) \\\dot{y} &= x^{2} + 3 y + 2 \end{aligned}\right. \) |
| 8. \( \left\{ \begin{aligned}\dot{x} &= - 4 x^{2} + y^{2} \\\dot{y} &= 4 y - 8 \end{aligned}\right. \) |
18. \( \left\{ \begin{aligned}\dot{x} &= x^{2} - y^{2} - 1 \\\dot{y} &= 2 y \end{aligned}\right. \) |
28. \( \left\{ \begin{aligned}\dot{x} &= - 3 x^{2} + x y + 12 x - 4 y \\\dot{y} &= x^{2} - 1 \end{aligned}\right. \) |
| 9. \( \left\{ \begin{aligned}\dot{x} &= \left(x - 3\right) \left(2 x - y\right) \\\dot{y} &= x y - 3 \end{aligned}\right. \) |
19. \( \left\{ \begin{aligned}\dot{x} &= - x^{2} - y^{2} + 1 \\\dot{y} &= 2 x \end{aligned}\right. \) |
29. \( \left\{ \begin{aligned}\dot{x} &= - 2 x^{2} + x y + 2 x - y \\\dot{y} &= \left(x + 3\right) \left(y - 1\right) \end{aligned}\right. \) |
| 10. \( \left\{ \begin{aligned}\dot{x} &= x^{2} - 6 x + y^{2} - 8 y \\\dot{y} &= x \left(- x + 2 y + 5\right) \end{aligned}\right. \) |
20. \( \left\{ \begin{aligned}\dot{x} &= x^{2} + y \\\dot{y} &= x^{2} - \left(y + 1\right)^{2} \end{aligned}\right. \) |
30. \( \left\{ \begin{aligned}\dot{x} &= \left(x + y\right) \left(y - 2\right) \\\dot{y} &= x - 1 \end{aligned}\right. \) |