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y"+y = 3 sin(2t) + t cos(2t) 2 mtupoy

y"+y = 3 sin(2t) + t cos(2t) 2 mtupoy

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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by 1. be convenient to treat these terms together, since each one individually may give rise to the sa If the nonhomogeneous function involves both cos(3t) and sin(t), it is usua form for a particular solution. For example, if g(t) = t sint+2 cost, the form for Y(t) wo ve the use of complex-valu aracteristic equation corresponding to the homogenec cquation, we must, of course, multiply each of the polynomials by t to increase their degro Y(t) = (Aot + A₁) sint +(Bot + B₁) cost, provided that sint and cost are not solutions of the homogeneous equation. Problems In each of Problems 1 through 10, find the general solution of the given differential equation. Xy" - 2y' - 3y = 3e²t 2._y" - y' - 2y = -2t+4t² 3. y"+y' - 6y = 12e³t + 12e-2t ve ALE 4. y" - 2y' - 3y = -3te-t 5. y"+2y' = 3 + 4 sin(2t) 6. y" +2y' + y = 2e-¹ X. y"+y = 3 sin(2t) + t cos(2t) ind 2 8. u" +wu = cos(wt), w² #w²/ 2 3. u" + w₁u = cos(wot) 10. y” +y+4y= 2 sinh t ni Hint: sinh t = (' - e-)/2 In each of Problems 11 through 15, find the solution of the given initial value problem. 11. y"+y'- 2y = 2t, y(0) = 0, y'(0) = 1 12. y" + 4y = 1² +3e", y(0) = 0, y'(0) = 2 13. y" - 2y + y = te' +4, y(0) = 1, y'(0) = 1 14. y" +4y= 3 sin(2t), y(0) = 2, y'(0) = -1 15. y" +2y + 5y = 4e In each of Problems 16 th Determine a su undetermined coeffi N b. Use a comput of the given equatio y" + 3y = 214 +t 16. 17. y" - 5y' +6y=e 18. y" +2y + 2y = 3 19. y" +4y= 12 sin(2 20. y" + 3y + 2y = e 21. y" +2y + 5y = 3 22. Consider the equati 13JAM y from Example 5. Recal solutions of the correspo method of reduction of nonhomogeneous equatio where v(t) is to be deter
Transcribed Image Text:by 1. be convenient to treat these terms together, since each one individually may give rise to the sa If the nonhomogeneous function involves both cos(3t) and sin(t), it is usua form for a particular solution. For example, if g(t) = t sint+2 cost, the form for Y(t) wo ve the use of complex-valu aracteristic equation corresponding to the homogenec cquation, we must, of course, multiply each of the polynomials by t to increase their degro Y(t) = (Aot + A₁) sint +(Bot + B₁) cost, provided that sint and cost are not solutions of the homogeneous equation. Problems In each of Problems 1 through 10, find the general solution of the given differential equation. Xy" - 2y' - 3y = 3e²t 2._y" - y' - 2y = -2t+4t² 3. y"+y' - 6y = 12e³t + 12e-2t ve ALE 4. y" - 2y' - 3y = -3te-t 5. y"+2y' = 3 + 4 sin(2t) 6. y" +2y' + y = 2e-¹ X. y"+y = 3 sin(2t) + t cos(2t) ind 2 8. u" +wu = cos(wt), w² #w²/ 2 3. u" + w₁u = cos(wot) 10. y” +y+4y= 2 sinh t ni Hint: sinh t = (' - e-)/2 In each of Problems 11 through 15, find the solution of the given initial value problem. 11. y"+y'- 2y = 2t, y(0) = 0, y'(0) = 1 12. y" + 4y = 1² +3e", y(0) = 0, y'(0) = 2 13. y" - 2y + y = te' +4, y(0) = 1, y'(0) = 1 14. y" +4y= 3 sin(2t), y(0) = 2, y'(0) = -1 15. y" +2y + 5y = 4e In each of Problems 16 th Determine a su undetermined coeffi N b. Use a comput of the given equatio y" + 3y = 214 +t 16. 17. y" - 5y' +6y=e 18. y" +2y + 2y = 3 19. y" +4y= 12 sin(2 20. y" + 3y + 2y = e 21. y" +2y + 5y = 3 22. Consider the equati 13JAM y from Example 5. Recal solutions of the correspo method of reduction of nonhomogeneous equatio where v(t) is to be deter
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