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10. v(dv/dx) = g; when x = xo,V %3D

10. v(dv/dx) = g; when x = xo,V %3D

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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Kindly answer item no. 10 and show your detailed solution. Thank you.
Sec. 71 Separation of Variables 25 The reader should note how the theorem of Section 6 applies to this problem to indicate that we have found implicitly the unique solution to the initial value problem which is continuous for y < -1. Exercises In Exercises 1 through 6, obtain the particular solution satisfying the initial condi- tion indicated. In each exercise interpret your answer in the light of the existence theorem of Section 6 and draw a graph of the solution. 1. dr/dt = - 4rt; when t = 0, r = ro - 2. 2xyy = 1 + y²; when x = 2, y = 3. 3. xyy' = 1 + y²; when x = 2, y = 3. 4. 2y dx = 3x dy; when x = 2, y = 1. 5. 2y dx = 3x dy; when x = –2, y = 1. 6. 2y dx = 3x dy; when x = 2, y = – 1. In Exercises 7 through 10, obtain the particular solution satisfying the initial condi- tion indicated. 7. y' = x exp (y – x²); when x = 0, y = 0. & xy2 dx + e dy = 0; when x → ∞, y → . 9. (2a? – r2) dr = r³ sin 0 de; when 0 = 0, r ='a. 10. v(dv/dx) = g; when x = X0,v = Vo . In Exercises 11 through 37, obtain the general solution. 11. (1 – x)y' = y². 12. sin x sin y dx + cos x cos y dy = 0. 13. xy dx + ez? dy = 0. 14. 2y dx = 3x dy. 15. my dx = nx dy. 16. y = xy². 17. dV/dP = – V/P. 18. ye2 dx = (4 + e2*) dy. 19. dr = b(cos 0 dr +r'sin 0 d0). 20. xy dx - (x + 2) dy = 0. 21. x2 dx + y(x – 1) dy = 0. 22. (xy + x) dx = (x²y² + x² + y? + 1) dy. 23. x cos? y dx + tan y dy = 0. 24. xy dx + (y + 1)e¯* dy = 0.
Transcribed Image Text:Sec. 71 Separation of Variables 25 The reader should note how the theorem of Section 6 applies to this problem to indicate that we have found implicitly the unique solution to the initial value problem which is continuous for y < -1. Exercises In Exercises 1 through 6, obtain the particular solution satisfying the initial condi- tion indicated. In each exercise interpret your answer in the light of the existence theorem of Section 6 and draw a graph of the solution. 1. dr/dt = - 4rt; when t = 0, r = ro - 2. 2xyy = 1 + y²; when x = 2, y = 3. 3. xyy' = 1 + y²; when x = 2, y = 3. 4. 2y dx = 3x dy; when x = 2, y = 1. 5. 2y dx = 3x dy; when x = –2, y = 1. 6. 2y dx = 3x dy; when x = 2, y = – 1. In Exercises 7 through 10, obtain the particular solution satisfying the initial condi- tion indicated. 7. y' = x exp (y – x²); when x = 0, y = 0. & xy2 dx + e dy = 0; when x → ∞, y → . 9. (2a? – r2) dr = r³ sin 0 de; when 0 = 0, r ='a. 10. v(dv/dx) = g; when x = X0,v = Vo . In Exercises 11 through 37, obtain the general solution. 11. (1 – x)y' = y². 12. sin x sin y dx + cos x cos y dy = 0. 13. xy dx + ez? dy = 0. 14. 2y dx = 3x dy. 15. my dx = nx dy. 16. y = xy². 17. dV/dP = – V/P. 18. ye2 dx = (4 + e2*) dy. 19. dr = b(cos 0 dr +r'sin 0 d0). 20. xy dx - (x + 2) dy = 0. 21. x2 dx + y(x – 1) dy = 0. 22. (xy + x) dx = (x²y² + x² + y? + 1) dy. 23. x cos? y dx + tan y dy = 0. 24. xy dx + (y + 1)e¯* dy = 0.
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