Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x.
1. dy/dx + 2xy = 0
2. dy/dx + 2xy^2 = 0
3. dy/dx = y sin x
4. (1 + x) dy/dx = 4y
5. 2√x dy/dx = √1 - y^2
6. dy/dx = 3√xy
7. dy/dx = (64xy)^(1/3)
8. dy/dx = 2x sec y
9. (1 - x^2) dy/dx = 2y
10. (1 + x)^2 dy/dx = (1 + y)^2
11. y' = xy^3
12. yy' = x(y^2 + 1)
13. y^3 dy/dx = (y^4 + 1) cos x
14. dy/dx = (1 + √x) / (1 + √y)
15. dy/dx = (x - 1)y^5 / (x^2(2y^3 - y))
16. (x^2 + 1)(tan y)y' = x
17. y' = 1 + x + y + xy (Suggestion: Factor the right-hand side.)
18. x^2y' = 1 - x^2 + y^2 - x^2y^2
Find explicit particular solutions of the initial value problems in Problems 19 through 28.
19. dy/dx = ye^x, y(0) = 2e
20. dy/dx = 3x^2(y^2 + 1), y(0) = 1
21. 2y dy/dx = x / √(x^2 - 16), y(5) = 2
22. dy/dx = 4x^3y - y, y(1) = -3
23. dy/dx + 1 = 2y, y(1) = 1
24. (tan x) dy/dx = y, y(1/2 π) = 1/2 π
25. x dy/dx - y = 2x^2y, y(1) = 1
26. dy/dx = 2xy^2 + 3x^2y^2, y(1) = -1
27. dy/dx = 6e^(2x-y), y(0) = 0