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13. 2y sin(xy) dx + (2x sin(xy) +3y²) dy = 0, y(0) = 1

13. 2y sin(xy) dx + (2x sin(xy) +3y²) dy = 0, y(0) = 1

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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Number 13 3.3 show all work
M = 3x +2y and ay 129 However, there is no function h(y) that would make this equal toN = x +xy, for were so that this equation is not exact. Suppose we try to solve this equation as we did in In Exercises 11-14, find the solution of the initial value N = 2x + y Example 1. We have -| Ms.9) dx = *y+; F (x, y) M(x, y) dx = y+xy +h(y). Next, F = ду 3 x² + 2xy +h'(y). this the case, + 2xy +h'(y) = x² + xy, which implies 1 h'(y) = - - xy. But this is impossible since the right-hand side depends on x and is not a function of y alone. We will see how to solve an equation such as this in Section 3.5. EXERCISES 3.3 13. 2y sin(xy) dx + (2x sin(xy) +3y²) dy = 0, In Exercises 1–10, determine if the differential equation is exact. If it is exact, find its solution. 1. (3x2 – 4y²) dx – (8xy – 12y³) dy = 0 2. (3xy+4y²) dx + (5x²y+2x²) dy = 0 3. (2xy+ ye*) dx + (x² + e*) dy = 0 4. (2xe +x²ye*y – 2) dx + x'e* dy = 0 5. (2x cos y – x²) dx + x² sin y dy = 0 y(0) = 1 14. | 1- dy + dx = 0, y(0) = 1 x² + y? x² + y? 15. Use Maple (or another appropriate software pack- age) to graph the solution in Exercise 11. 16. Use Maple (or another appropriate software pack- age) to graph the solution in Exercise 12. 6. (y cos x+3e* cos y) dx+(sin x – 3e* sin y) dy = 0 x² – 2xy + 1 x² – y? 18. Show the converse of Theorem 3.2 can be proved by mo integrating with respect to y first. 17. Show a separable differential equation is exact. 1- 2xy 1. y = x² 8. y' = 19. Determine conditions on a, b, c, and d so that the differential equation 0 -r cos 0 ds e'-2t cos s 9. - = dt dr 10. do es -12 sin s r + sin 0 ax + by y' cx +dy is exact and, for a differential equation satisfying these conditions, solve the differential equation. problem. 11. 2ry dx + (x² + 3y²) dy = 0, y(1) = 1 12. y = 2re - 3x²y x3 - x2ey , y(1) = 0
Transcribed Image Text:M = 3x +2y and ay 129 However, there is no function h(y) that would make this equal toN = x +xy, for were so that this equation is not exact. Suppose we try to solve this equation as we did in In Exercises 11-14, find the solution of the initial value N = 2x + y Example 1. We have -| Ms.9) dx = *y+; F (x, y) M(x, y) dx = y+xy +h(y). Next, F = ду 3 x² + 2xy +h'(y). this the case, + 2xy +h'(y) = x² + xy, which implies 1 h'(y) = - - xy. But this is impossible since the right-hand side depends on x and is not a function of y alone. We will see how to solve an equation such as this in Section 3.5. EXERCISES 3.3 13. 2y sin(xy) dx + (2x sin(xy) +3y²) dy = 0, In Exercises 1–10, determine if the differential equation is exact. If it is exact, find its solution. 1. (3x2 – 4y²) dx – (8xy – 12y³) dy = 0 2. (3xy+4y²) dx + (5x²y+2x²) dy = 0 3. (2xy+ ye*) dx + (x² + e*) dy = 0 4. (2xe +x²ye*y – 2) dx + x'e* dy = 0 5. (2x cos y – x²) dx + x² sin y dy = 0 y(0) = 1 14. | 1- dy + dx = 0, y(0) = 1 x² + y? x² + y? 15. Use Maple (or another appropriate software pack- age) to graph the solution in Exercise 11. 16. Use Maple (or another appropriate software pack- age) to graph the solution in Exercise 12. 6. (y cos x+3e* cos y) dx+(sin x – 3e* sin y) dy = 0 x² – 2xy + 1 x² – y? 18. Show the converse of Theorem 3.2 can be proved by mo integrating with respect to y first. 17. Show a separable differential equation is exact. 1- 2xy 1. y = x² 8. y' = 19. Determine conditions on a, b, c, and d so that the differential equation 0 -r cos 0 ds e'-2t cos s 9. - = dt dr 10. do es -12 sin s r + sin 0 ax + by y' cx +dy is exact and, for a differential equation satisfying these conditions, solve the differential equation. problem. 11. 2ry dx + (x² + 3y²) dy = 0, y(1) = 1 12. y = 2re - 3x²y x3 - x2ey , y(1) = 0
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