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Exact Solutions > Ordinary Differential Equations > First-Order Ordinary Differential Equations

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## 1. First-Order Ordinary Differential Equations

1. y = f(y).    Autonomous equation.
2. y = f(x)g(y).    Separable equation.
3. g(x)y = f1(x)y + f0(x).    Linear equation.
4. g(x)y = f1(x)y + fn(x)yn.    Bernoulli equation.
5. y = f(y/x).    Homogeneous equation
6. y = ay2 + bxn.    Special Riccati equation.
7. y = y2 + f(x)ya2af(x).    Riccati equation, special case 1.
8. y = f(x)y2 + ayabb2f(x).    Riccati equation, special case 2.
9. y = y2 + xf(x)y + f(x).    Riccati equation, special case 3.
10. y = f(x)y2axnf(x)y + anxn−1.    Riccati equation, special case 4.
11. y = f(x)y2 + anxn−1a2x2nf(x).    Riccati equation, special case 5.
12. y = −(n + 1)xny2 + xn+1f(x)yf(x).    Riccati equation, special case 6.
13. xy = f(x)y2 + ny + ax2nf(x).    Riccati equation, special case 7.
14. xy = x2nf(x)y2 + [axnf(x) − n]y + bf(x).    Riccati equation, special case 8.
15. y = f(x)y2 + g(x)ya2f(x) − ag(x).    Riccati equation, special case 9.
16. y = f(x)y2 + g(x)y + anxn−1a2x2nf(x) − axng(x).    Riccati equation, special case 10.
17. y = aeλxy2 + aeλxf(x)y + λf(x).    Riccati equation, special case 11.
18. y = f(x)y2aeλxf(x)y + aλeλx.    Riccati equation, special case 12.
19. y = f(x)y2 + aλeλxa2e2λxf(x).    Riccati equation, special case 13.
20. y = f(x)y2 + λy + ae2λxf(x).    Riccati equation, special case 14.
21. y = y2f2(x) + f(x).    Riccati equation, special case 15.
22. y = f(x)y2f(x)g(x)y + g(x).    Riccati equation, special case 16.
23. y = f(x)y2 + g(x)y + h(x).    General Riccati equation.
24. yy = y + f(x).    Abel equation of the second kind in the canonical form.
25. yy = f(x)y + g(x).    Abel equation of the second kind.
26. yy = f(x)y2 + g(x)y + h(x).    Abel equation of the second kind.
27. y = f(ax + by + c).
28. y = f(y + axn + b) − anxn−1.
29. y = (y/x)f(xnym).    Generalized homogeneous equation.
30. y = −(n/m)(y/x) + ykf(x)g(xnym).
31. y = f((ax + by + c)/(αx + βy + γ)).
32. y = xn−1y1−mf(axn + bym).
33. [xnf(y) + xg(y)]y = h(y).
34. x[f(xnym) + mxkg(xnym)]y = y[h(xnym) − nxkg(xnym)].
35. x[f(xnym) + mykg(xnym)]y = y[h(xnym) − nykg(xnym)].
36. x[sf(xnym) − mg(xkys)]y = y[ng(xkys) − kf(xnym)].
37. [f(y) + amxnym−1]y + g(x) + anxn−1ym = 0.
38. y = eλxf(eλxy).
39. y = eλyf(eλyx).
40. y = yf(eαxym).
41. y = x−1f(xneαy).
42. y = f(x)eλy + g(x).
43. y = −nx−1 + f(x)g(xney).
44. y = −(α/m)y + ykf(x)g(eαxym).
45. y = eαxβyf(aeαx + beβy).
46. [eαxf(y) + ]y + eβyg(x) + = 0.
47. x[f(xneαy) + αyg(xneαy)]y = h(xneαy) − nyg(xneαy).
48. [f(eαxym) + mxg(eαxym)]y = y[h(eαxym) − αxg(eαxym)].

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