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In each of Problems 1 through 10, find the general solution of the given differential equation. TO Xy" - 2y' - 3y = 3e²t

In each of Problems 1 through 10, find the general solution of the given differential equation. TO Xy" - 2y' - 3y = 3e²t

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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gree nial that a e a CDB e by 1. Y(t) = eat (Aot" +...+An) cos(ßt) + eat (Both +...+ Bn) sin(ßt). coefficients. If at iß satisfy the characteristic equation corresponding to the homogeneou Usually, the latter form is preferred because it does not involve the use of complex-valuec equation, we must, of course, multiply each of the polynomials by t to increase their degree If the nonhomogeneous function involves both cos(3t) and sin(3t), it is usuall convenient to treat these terms together, since each one individually may give rise to the sam form for a particular solution. For example, if g(t) = t sin t+2 cost, the form for Y(t) woul be or, equiva Y(t) = (Aot + A₁) sint +(Bot + B₁) cost, provided that sint and cost are not solutions of the homogeneous equation. Xy" - 2y' - 3y = 3e²t 2. y"-y' - 2y = -2t+41² 3. y"+y'-6y = 12e³ + 12e-2 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) Problems ener In each of Problems 1 through 10, find the general solution of the given differential equation. 2 8. u" + wu = cos(wt), w² #wo 09251 holbal to 29011011 )424 + Bn), 2 u" + 3. u" + wou wu = cos(wot) ni 10. y” + y+4y = 2 sinht Hint: sinht = (e! – 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 thr Determine a sui undetermined coeffic N b. Use a compute of the given equation 16. y" + 3y' = 2t4 +1² 17. y" - 5y +6y=e¹ 18. y" +2y + 2y = 3e 19. y" +4y= 1² sin(21 20. y" + 3y + 2y = ef 21. y" +2y + 5y = 3a 22. Consider the equati y from Example 5. Recall solutions of the correspa method of reduction of nonhomogeneous equatic where v(t) is to be deter
Transcribed Image Text:gree nial that a e a CDB e by 1. Y(t) = eat (Aot" +...+An) cos(ßt) + eat (Both +...+ Bn) sin(ßt). coefficients. If at iß satisfy the characteristic equation corresponding to the homogeneou Usually, the latter form is preferred because it does not involve the use of complex-valuec equation, we must, of course, multiply each of the polynomials by t to increase their degree If the nonhomogeneous function involves both cos(3t) and sin(3t), it is usuall convenient to treat these terms together, since each one individually may give rise to the sam form for a particular solution. For example, if g(t) = t sin t+2 cost, the form for Y(t) woul be or, equiva Y(t) = (Aot + A₁) sint +(Bot + B₁) cost, provided that sint and cost are not solutions of the homogeneous equation. Xy" - 2y' - 3y = 3e²t 2. y"-y' - 2y = -2t+41² 3. y"+y'-6y = 12e³ + 12e-2 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) Problems ener In each of Problems 1 through 10, find the general solution of the given differential equation. 2 8. u" + wu = cos(wt), w² #wo 09251 holbal to 29011011 )424 + Bn), 2 u" + 3. u" + wou wu = cos(wot) ni 10. y” + y+4y = 2 sinht Hint: sinht = (e! – 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 thr Determine a sui undetermined coeffic N b. Use a compute of the given equation 16. y" + 3y' = 2t4 +1² 17. y" - 5y +6y=e¹ 18. y" +2y + 2y = 3e 19. y" +4y= 1² sin(21 20. y" + 3y + 2y = ef 21. y" +2y + 5y = 3a 22. Consider the equati y from Example 5. Recall solutions of the correspa method of reduction of nonhomogeneous equatic where v(t) is to be deter
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