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18. x²y' = 1 – x² + y² – x²y²

18. x²y' = 1 – x² + y² – x²y²

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1.4 Problems Find general solutions (implicit if necessary, explicit if conve- nient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. dy dy 1. + 2xy = 0 2. + 2xy² = 0 dx dx dy dy 3. = y sin x 4. (1+x) = 4y dx dx dy 5. – y? 6. = 3/xy dx dx dy 7. 3 (64ху)\в dx dy 8. = 2x sec y dx 9. (1 – x²) = 2y 10. (1+x)² dx = (1 + y)? dx 11. у %3 ху 12. yy' = x(y² + 1) dy _1+ a 1+ y dy 13. y = (y* + 1) cos x 14. %3D dx dx dy (x – 1)y³ 15. dx x²(2y³ – y) 16. (x² + 1)(tan y)y' = x 17. y' = 1+x+y+xy (Suggestion: Factor the right-hand side.) 18. x*y = 1-x²+ y? – x²y? Find explicit particular solutions of the initial value problems in Problems 19 through 28. dy 19. = ye", y(0) = 2e dx dy 20. = 3x²(y² + 1), y(0) = 1 dx dy 21. 2y dx y(5) = 2 Vx? – 16 dy 22. = 4x³y – y. y(1) = -3 dx dy 23. dx +1 = 2y, y(1) = 1 dy 24. (tan x)- - = y. y (7) = }7 dx dy 25. x- y = 2x²y, y(1) = 1 dy 26. = 2xy² + 3x²y², y(1) = -1 dx dy 27. = 6e²x-y, y(0) =0 %3D dx 28. 2 dx - cos y, y(4) = 1/4 29. (a) Find a general solution of the differential equation dy/dx = y. (b) Find a singular solution that is not in- cluded in the general solution. (c) Inspect a sketch of typi- cal solution curves to determine the points (a, b)for which the initial value problem y = y', y(a) = b has a unique solution.
Transcribed Image Text:1.4 Problems Find general solutions (implicit if necessary, explicit if conve- nient) of the differential equations in Problems 1 through 18. Primes denote derivatives with respect to x. dy dy 1. + 2xy = 0 2. + 2xy² = 0 dx dx dy dy 3. = y sin x 4. (1+x) = 4y dx dx dy 5. – y? 6. = 3/xy dx dx dy 7. 3 (64ху)\в dx dy 8. = 2x sec y dx 9. (1 – x²) = 2y 10. (1+x)² dx = (1 + y)? dx 11. у %3 ху 12. yy' = x(y² + 1) dy _1+ a 1+ y dy 13. y = (y* + 1) cos x 14. %3D dx dx dy (x – 1)y³ 15. dx x²(2y³ – y) 16. (x² + 1)(tan y)y' = x 17. y' = 1+x+y+xy (Suggestion: Factor the right-hand side.) 18. x*y = 1-x²+ y? – x²y? Find explicit particular solutions of the initial value problems in Problems 19 through 28. dy 19. = ye", y(0) = 2e dx dy 20. = 3x²(y² + 1), y(0) = 1 dx dy 21. 2y dx y(5) = 2 Vx? – 16 dy 22. = 4x³y – y. y(1) = -3 dx dy 23. dx +1 = 2y, y(1) = 1 dy 24. (tan x)- - = y. y (7) = }7 dx dy 25. x- y = 2x²y, y(1) = 1 dy 26. = 2xy² + 3x²y², y(1) = -1 dx dy 27. = 6e²x-y, y(0) =0 %3D dx 28. 2 dx - cos y, y(4) = 1/4 29. (a) Find a general solution of the differential equation dy/dx = y. (b) Find a singular solution that is not in- cluded in the general solution. (c) Inspect a sketch of typi- cal solution curves to determine the points (a, b)for which the initial value problem y = y', y(a) = b has a unique solution.
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