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(x² + y2)3³/2 (x² + y2)3/2dx In each of Problems 9 and 10, solve the given initial value problem and determine at least approximately where the solution is valid. 9. (2x - y) + (2y-x) y' = 0, y(1) = 3 10. (9r²+1 1) (^ xx-0 (1)-0 ha In

(x² + y2)3³/2 (x² + y2)3/2dx In each of Problems 9 and 10, solve the given initial value problem and determine at least approximately where the solution is valid. 9. (2x - y) + (2y-x) y' = 0, y(1) = 3 10. (9r²+1 1) (^ xx-0 (1)-0 ha In

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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Problems Determine whether each of the equations in Problems 1 through 8 is exact. If it is exact, find the solution. 19,03 1. (2x+3)+(2y = 2)y' = 0 m (2x+4y) + (2x - 2y) y' =0 4. 2. 3. (3x² - 2xy + 2) + (6y² − x² + 3) y' = 0 dy ax + by dx bx + cy ax - by bx - cy 5. dy dx 8. = shit Shy hot Bishfuldshinwoner og Sigma 10-1 S substituting it into the tangent line approximion at i = 6. (yety cos(2x) -2exy sin(2x)+2x)+(xey cos(2x)-3) y' = 0 (y/x+6x) + (lnx - 2) y' = 0, x > 0 7. X y dy (x² + y²) 3/2 + (x² + y²) 3/2 dx = S where (xo, yo) is a point in R. dod sdi svilauziy no voy bibit noitoni 20 gninate noitulos s idi. T is also exact. nodi bre sulemami 7-20+% Y- In each of Problems 9 and 10, solve the given initial value problem and determine at least approximately where the solution is valid. y(1) = 3 9. (2x - y) + (2y - x) y' = 0, 10. (9x² + y-1) - (4y - x) y' = 0, y(1) = 0 (x, y) = = 0 In each of Problems 11 and 12, find the value of b for which the given equation is exact, and then solve it using that value of b. YEAH 11. (xy² + bx²y) + (x+y)x²y' = 0 12. (ye²xy + x) + bxe²xy y' = 0 13. Assume that equation (6) meets the requirements of Theorem 2.6.1 in a rectangle R and is therefore exact. Show that a possible function (x, y) is M(s, yo) ds + Yo 14. Show that any separable equation N(x, t) dt, M(x) + N(y) y' = 0 In each of Problems 15 and 16, show that the given equation is not exact but becomes exact when multiplied by the given integrating factor. Then solve the equation. 15. x²y³ + x(1+ y²) y' = 0, p(x, y) = 1/(xy³) 16. (x+2) sin y + (x cos y) y' = 0, μ(x, y) = xe* 17. Show that if (Nx - My)/M = Q, where Q is a function of y only, then the differential equation M + Ny' = 0 has an integrating factor of the form μ(y) = exp [ Q(x)dy. In each of Problems 18 through 21, find an integrating factor and solve the given equation. 18. (3x²y + 2xy + y³) + (x² + y²) y' = 0 19. y' = e²x + y - 1 20. 1+(x/y- sin y) y' = 0 21. y + (2xy-e-2y) y' = 0 22. Solve the differential equation (3xy + y²) + (x² + xy) y' = 0 using the integrating factor u(x, y) = (xy(2x + y))-¹. Verify that the solution is the same as that obtained in Example 4 with a different integrating factor.
Transcribed Image Text:Problems Determine whether each of the equations in Problems 1 through 8 is exact. If it is exact, find the solution. 19,03 1. (2x+3)+(2y = 2)y' = 0 m (2x+4y) + (2x - 2y) y' =0 4. 2. 3. (3x² - 2xy + 2) + (6y² − x² + 3) y' = 0 dy ax + by dx bx + cy ax - by bx - cy 5. dy dx 8. = shit Shy hot Bishfuldshinwoner og Sigma 10-1 S substituting it into the tangent line approximion at i = 6. (yety cos(2x) -2exy sin(2x)+2x)+(xey cos(2x)-3) y' = 0 (y/x+6x) + (lnx - 2) y' = 0, x > 0 7. X y dy (x² + y²) 3/2 + (x² + y²) 3/2 dx = S where (xo, yo) is a point in R. dod sdi svilauziy no voy bibit noitoni 20 gninate noitulos s idi. T is also exact. nodi bre sulemami 7-20+% Y- In each of Problems 9 and 10, solve the given initial value problem and determine at least approximately where the solution is valid. y(1) = 3 9. (2x - y) + (2y - x) y' = 0, 10. (9x² + y-1) - (4y - x) y' = 0, y(1) = 0 (x, y) = = 0 In each of Problems 11 and 12, find the value of b for which the given equation is exact, and then solve it using that value of b. YEAH 11. (xy² + bx²y) + (x+y)x²y' = 0 12. (ye²xy + x) + bxe²xy y' = 0 13. Assume that equation (6) meets the requirements of Theorem 2.6.1 in a rectangle R and is therefore exact. Show that a possible function (x, y) is M(s, yo) ds + Yo 14. Show that any separable equation N(x, t) dt, M(x) + N(y) y' = 0 In each of Problems 15 and 16, show that the given equation is not exact but becomes exact when multiplied by the given integrating factor. Then solve the equation. 15. x²y³ + x(1+ y²) y' = 0, p(x, y) = 1/(xy³) 16. (x+2) sin y + (x cos y) y' = 0, μ(x, y) = xe* 17. Show that if (Nx - My)/M = Q, where Q is a function of y only, then the differential equation M + Ny' = 0 has an integrating factor of the form μ(y) = exp [ Q(x)dy. In each of Problems 18 through 21, find an integrating factor and solve the given equation. 18. (3x²y + 2xy + y³) + (x² + y²) y' = 0 19. y' = e²x + y - 1 20. 1+(x/y- sin y) y' = 0 21. y + (2xy-e-2y) y' = 0 22. Solve the differential equation (3xy + y²) + (x² + xy) y' = 0 using the integrating factor u(x, y) = (xy(2x + y))-¹. Verify that the solution is the same as that obtained in Example 4 with a different integrating factor.
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