1.5 Problems Find general solutions of the differential equations in Prob- lems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. 21. xy = 3y +x*cos x, y(27) = 0 22. y = 2xy + 3x² exp(x²), y(0) = 5 23. ху + (2х — 3)у 3 4x* 24. (x² + 4)y' + 3xy = x, y(0) = 1 dy 1. у' +у3 2, у (0)- 0 2. y' – 2y = 3e2", y (0) = 0 3. у + 3у 3 2хе-м 4. у — 2ху %3е 5. xy +2y = 3x, y(1) = 5 6. xy' + 5y = 7x², y(2) = 5 7. 2xy' + y = 10E 8. 3xy' + y = 12x 9. ху' — у — х, у(1) - 7 10. 2ху' — Зу - 9х3 11. ху + у3 Зху, у(1) - 0 12. ху + 3у 3D 2х', у (2) — 1 13. у +у-е', у(0) — 1 14. ху — Зу %3D х*, у(1) —D 10 15. y + 2xy = x, y(0) = -2 16. y'%3 (1 — у) сos x, у(л) 2 17. (1 + х)у' + у 3 сos.x, у (0) %3 1 18. xy' = 2y +x' cosx 19. y + y cot x = cos x 20. у 3D1+х + у+ху, у (0) 3D0 25. (x² + 1) + 3x³y = 6x exp (–}x²), y(0) =1 Solve the differential equations in Problems 26 through 28 by regarding y as the independent variable rather than x. 26. (1 - 4ху?) - у dx dy 27. (x + ye') dy = 1 dx dy 28. (1+ 2xy) = 1+ y? 29. Express the general solution of dy/dx =1+2xy in terms of the error function erf(x) 30. Express the solution of the initial value problem 2x = y + 2x cos x, y(1) =0 as an integral as in Example 3 of this section.
1.5 Problems Find general solutions of the differential equations in Prob- lems 1 through 25. If an initial condition is given, find the corresponding particular solution. Throughout, primes denote derivatives with respect to x. 21. xy = 3y +x*cos x, y(27) = 0 22. y = 2xy + 3x² exp(x²), y(0) = 5 23. ху + (2х — 3)у 3 4x* 24. (x² + 4)y' + 3xy = x, y(0) = 1 dy 1. у' +у3 2, у (0)- 0 2. y' – 2y = 3e2", y (0) = 0 3. у + 3у 3 2хе-м 4. у — 2ху %3е 5. xy +2y = 3x, y(1) = 5 6. xy' + 5y = 7x², y(2) = 5 7. 2xy' + y = 10E 8. 3xy' + y = 12x 9. ху' — у — х, у(1) - 7 10. 2ху' — Зу - 9х3 11. ху + у3 Зху, у(1) - 0 12. ху + 3у 3D 2х', у (2) — 1 13. у +у-е', у(0) — 1 14. ху — Зу %3D х*, у(1) —D 10 15. y + 2xy = x, y(0) = -2 16. y'%3 (1 — у) сos x, у(л) 2 17. (1 + х)у' + у 3 сos.x, у (0) %3 1 18. xy' = 2y +x' cosx 19. y + y cot x = cos x 20. у 3D1+х + у+ху, у (0) 3D0 25. (x² + 1) + 3x³y = 6x exp (–}x²), y(0) =1 Solve the differential equations in Problems 26 through 28 by regarding y as the independent variable rather than x. 26. (1 - 4ху?) - у dx dy 27. (x + ye') dy = 1 dx dy 28. (1+ 2xy) = 1+ y? 29. Express the general solution of dy/dx =1+2xy in terms of the error function erf(x) 30. Express the solution of the initial value problem 2x = y + 2x cos x, y(1) =0 as an integral as in Example 3 of this section.
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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Transcribed Image Text:1.5 Problems
Find general solutions of the differential equations in Prob-
lems 1 through 25. If an initial condition is given, find the
corresponding particular solution. Throughout, primes denote
derivatives with respect to x.
21. xy = 3y +x*cos x, y(27) = 0
22. y = 2xy + 3x² exp(x²), y(0) = 5
23. ху + (2х — 3)у 3 4x*
24. (x² + 4)y' + 3xy = x, y(0) = 1
dy
1. у' +у3 2, у (0)- 0
2. y' – 2y = 3e2", y (0) = 0
3. у + 3у 3 2хе-м
4. у — 2ху %3е
5. xy +2y = 3x, y(1) = 5
6. xy' + 5y = 7x², y(2) = 5
7. 2xy' + y = 10E
8. 3xy' + y = 12x
9. ху' — у — х, у(1) - 7
10. 2ху' — Зу - 9х3
11. ху + у3 Зху, у(1) - 0
12. ху + 3у 3D 2х', у (2) — 1
13. у +у-е', у(0) — 1
14. ху — Зу %3D х*, у(1) —D 10
15. y + 2xy = x, y(0) = -2
16. y'%3 (1 — у) сos x, у(л) 2
17. (1 + х)у' + у 3 сos.x, у (0) %3 1
18. xy' = 2y +x' cosx
19. y + y cot x = cos x
20. у 3D1+х + у+ху, у (0) 3D0
25. (x² + 1) + 3x³y = 6x exp (–}x²), y(0) =1
Solve the differential equations in Problems 26 through 28 by
regarding y as the independent variable rather than x.
26. (1 - 4ху?) - у
dx
dy
27. (x + ye')
dy
= 1
dx
dy
28. (1+ 2xy) = 1+ y?
29. Express the general solution of dy/dx =1+2xy in terms
of the error function
erf(x)
30. Express the solution of the initial value problem
2x = y + 2x cos x, y(1) =0
as an integral as in Example 3 of this section.
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