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18 through 21, find an integrating factor and solve In each of Problems 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

18 through 21, find an integrating factor and solve In each of Problems 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

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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y' = 0 oblem 1. given prem 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, 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 exp foly 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 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:y' = 0 oblem 1. given prem 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, 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 exp foly 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 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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