9.3 Exercises In Exercises 1-59, find a particular solution. 1. 2-2y-5y+6y=e-332-23x+6x^2 2. 3.4y+8y-2y=-e4+45x+9x^2 3. y+3y-y-3y=e-22-17x+3x 4. +3y--3y=e-1+2x+24x+16x 5. y+y-2y=e14+34x+15x 6. 4y+8y-y-2y=-e-21-15x 7. y=y-y+y=e7+6x x08+12=f+f+f-nfZ6 8. 12.8y12y+6y-y=e/21+4x 9. y4+3y-3y-7y+6y=-e-12+8x-8x 10. 2I+I28-=hf8-f+ufg+mf 11. y(4+8y+24y+32=-16e-2x1+x+x2-x 12. xg-1-=2f6-f1I-n 13. y4=2y+3y-y=e3+4x+x 14. +x+2x^2=f2+f-fg+f= 15. 2x1+x82+2-=fp-f/gf-uf+ 16. 2y(4+y-2y=y=3c-x/21-6x 17. y-5y'+4y=c3+x-3x 18. y4-2y-3y+4y+4y=e213+33x+18x 19. y-3y+4y/=e215+26x+12x 20. y4-2y+2y-y=e1+x) (221+I1=f-f+fg+ufg-i9
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2-2y-5y+6y=e-332-23x+6x^2 Simplifying the left side, we have: 2-2y-5y+6y = 2-y Show more…
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In Problems 1-26 solve the given differential equation by undetermined coefficients. 1. y'' + 3y' + 2y = 6 2. 4y'' + 9y = 15 3. y'' - 10y' + 25y = 30x + 3 4. y'' + y' - 6y = 2x 5. 1/4y'' + y' + y = x^2 - 2x 6. y'' - 8y' + 20y = 100x^2 - 26xe^x 7. y'' + 3y = -48x^2e^{3x} 8. 4y'' - 4y' - 3y = cos 2x 9. y'' - y' = -3 10. y'' + 2y' = 2x + 5 - e^{-2x} 11. y'' - y' + 1/4y = 3 + e^{x/2} 12. y'' - 16y = 2e^{4x} 13. y'' + 4y = 3 sin 2x 14. y'' - 4y = (x^2 - 3) sin 2x 15. y'' + y = 2x sin x 16. y'' - 5y' = 2x^3 - 4x^2 - x + 6 17. y'' - 2y' + 5y = e^x cos 2x 18. y'' - 2y' + 2y = e^{2x}(cos x - 3 sin x) 19. y'' + 2y' + y = sin x + 3 cos 2x 20. y'' + 2y' - 24y = 16 - (x + 2)e^{4x} 21. y''' - 6y'' = 3 - cos x 22. y''' - 2y'' - 4y' + 8y = 6xe^{2x} 23. y''' - 3y'' + 3y' - y = x - 4e^x 24. y''' - y'' - 4y' + 4y = 5 - e^x + e^{2x} 25. y^{(4)} + 2y'' + y = (x - 1)^2 26. y^{(4)} - y'' = 4x + 2xe^{-x} In Problems 27-36 solve the given initial-value problems. 27. y'' + 4y = -2, y(π/8) = 1/2, y'(π/8) = 2 28. 2y'' + 3y' - 2y = 14x^2 - 4x - 11, y(0) = 0, y'(0) = 0 29. 5y'' + y' = -6x, y(0) = 0, y'(0) = -10 30. y'' + 4y' + 4y = (3 + x)e^{-2x}, y(0) = 2, y'(0) = 0 31. y'' + 4y' + 5y = 35e^{-4x}, y(0) = -3, y'(0) = 1
Adi S.
In Problems 1-14 find the general solution of the given second-order differential equation. 1. 4y'' + y' = 0 2. y'' - 36y = 0 3. y'' - y' - 6y = 0 4. y'' - 3y' + 2y = 0 5. y'' + 8y' + 16y = 0 6. y'' - 10y' + 25y = 0 7. 12y'' - 5y' - 2y = 0 8. y'' + 4y' - y = 0 9. y'' + 9y = 0 10. 3y'' + y = 0 11. y'' - 4y' + 5y = 0 12. 2y'' + 2y' + y = 0 13. 3y'' + 2y' + y = 0 14. 2y'' - 3y' + 4y = 0 In Problems 15-28 find the general solution of the given higher-order differential equation. 15. y''' - 4y'' - 5y' = 0 16. y''' - y = 0 17. y''' - 5y'' + 3y' + 9y = 0 18. y''' + 3y'' - 4y' - 12y = 0 19. d^3u/dt^3 + d^2u/dt^2 - 2u = 0 20. d^3x/dt^3 - d^2x/dt^2 - 4x = 0 21. y''' + 3y'' + 3y' + y = 0 22. y''' - 6y'' + 12y' - 8y = 0 23. y^(4) + y''' + y'' = 0 24. y^(4) - 2y'' + y = 0 25. 16 d^4y/dx^4 + 24 d^2y/dx^2 + 9y = 0 26. d^4y/dx^4 - 7 d^2y/dx^2 - 18y = 0 27. d^5u/dr^5 + 5 d^4u/dr^4 - 2 d^3u/dr^3 - 10 d^2u/dr^2 + du/dr + 5u = 0 28. 2 d^5x/ds^5 - 7 d^4x/ds^4 + 12 d^3x/ds^3 + 8 d^2x/ds^2 = 0 In Problems 29-36 solve the given initial-value problem. 29. y'' + 16y = 0, y(0) = 2, y'(0) = -2 30. d^2y/dθ^2 + y = 0, y(π/3) = 0, y'(π/3) = 2
Sri K.
1.6 Problems Find general solutions of the differential equations in Problems 1 through 30. Primes denote derivatives with respect to x throughout. 1. (x + y)y' = x - y 2. 2xyy' = x^2 + 2y^2 3. xy' = y + 2√(xy) 4. (x - y)y' = x + y 5. x(x + y)y' = y(x - y) 6. (x + 2y)y' = y 7. xy^2y' = x^3 + y^3 8. x^2y' = xy + x^2e^{y/x} 9. x^2y' = xy + y^2 10. xyy' = x^2 + 3y^2 11. (x^2 - y^2)y' = 2xy 12. xyy' = y^2 + x√(4x^2 + y^2) 13. xy' = y + √(x^2 + y^2) 14. yy' + x = √(x^2 + y^2) 15. x(x + y)y' + y(3x + y) = 0 16. y' = √(x + y + 1) 17. y' = (4x + y)^2 18. (x + y)y' = 1 19. x^2y' + 2xy = 5y^3 20. y^2y' + 2xy^3 = 6x 21. y' = y + y^3 22. x^2y' + 2xy = 5y^4 23. xy' + 6y = 3xy^{4/3} 24. 2xy' + y^3e^{-2x} = 2xy 25. y^2(xy' + y)(1 + x^4)^{1/2} = x 26. 3y^2y' + y^3 = e^{-x} 27. 3xy^2y' = 3x^4 + y^3 28. xe^yy' = 2(e^y + x^3e^{2x}) 29. (2x sin y cos y)y' = 4x^2 + sin^2 y 30. (x + e^x)y' = xe^{-y} - 1 In Problems 31 through 42, verify that the given differential equation is exact; then solve it. 31. (2x + 3y) dx + (3x + 2y) dy = 0 32. (4x - y) dx + (6y - x) dy = 0 33. (3x^2 + 2y^2) dx + (4xy + 6y^2) dy = 0 34. (2xy^2 + 3x^2) dx + (2x^2y + 4y^3) dy = 0 35. (x^3 + y/x) dx + (y^2 + ln x) dy = 0 36. (1 + ye^{xy}) dx + (2y + xe^{xy}) dy = 0 37. (cos x + ln y) dx + (x/y + e^y) dy = 0 38. (x + tan^{-1} y) dx + (x + y)/(1 + y^2) dy = 0 39. (3x^2y^3 + y^4) dx + (3x^3y^2 + y^4 + 4xy^3) dy = 0 40. (e^x sin y + tan y) dx + (e^x cos y + x sec^2 y) dy = 0 41. (2x/y - 3y^2/x^4) dx + (2y/x^3 - x^2/y^2 + 1/√(y)) dy = 0 42. (2x^{5/2} - 3y^{5/3})/(2x^{5/2}y^{2/3}) dx + (3y^{5/3} - 2x^{5/2})/(3x^{3/2}y^{5/3}) dy = 0 Find a general solution of each reducible second-order differential equation in Problems 43-54. Assume x, y and/or y' positive where helpful (as in Example 11). 43. xy'' = y' 44. yy'' + (y')^2 = 0 45. y'' + 4y = 0 46. xy'' + y' = 4x 47. y'' = (y')^2 48. x^2y'' + 3xy' = 2 49. yy'' + (y')^2 = yy' 50. y'' = (x + y')^2 51. y'' = 2y(y')^3 52. y^3y'' = 1 53. y'' = 2yy' 54. yy'' = 3(y')^2 55. Show that the substitution v = ax + by + c transforms the differential equation dy/dx = F(ax + by + c) into a separable equation. 56. Suppose that n ≠ 0 and n ≠ 1. Show that the substitution v = y^{1-n} transforms the Bernoulli equation dy/dx + P(x)y = Q(x)y^n into the linear equation dv/dx + (1 - n)P(x)v(x) = (1 - n)Q(x).
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