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rough 8 is 14. Show that any separable equation ) y' = 0 roblem d. given brem sible M(x) + N(y) y' = 0 is also exact. 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, u(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 exp / 003 μ(ν) = exp Q(y)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 (x, y) = (xy(2x + y))-¹. Verify that the solution is the same as that obtained in Example 4 with a different integrating factor.

rough 8 is 14. Show that any separable equation ) y' = 0 roblem d. given brem sible M(x) + N(y) y' = 0 is also exact. 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, u(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 exp / 003 μ(ν) = exp Q(y)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 (x, y) = (xy(2x + y))-¹. Verify that the solution is the same as that obtained in Example 4 with a different integrating factor.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 8CR
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rough 8 is )y' = 0 roblem d. given brem sible 14. Show that any separable equation M(x) + N(y) y' = 0 is also exact. 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, u(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 Pfer μ(y) = exp Q(y)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 µ(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:rough 8 is )y' = 0 roblem d. given brem sible 14. Show that any separable equation M(x) + N(y) y' = 0 is also exact. 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, u(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 Pfer μ(y) = exp Q(y)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 µ(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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