15. x(x + y)y' + y(3x + y) = 0 16. y = Vx+y+T 18. (x + y)y = 1 20. y'y +2xy = 6x 22. x'y +2xy = 5y 24. 2xy + y'e-2 = 2xy 25. y(xy + y)(1 +x*)2 =x 26. 3y?y' +y =e* 27. 3xy'y 3x + y 28. xe'y = 2(e +x'e) 29. (2x sin y cos y)y 4x + sin' y 30. (x +e')y' = xe -1 Find general solutions of the differential equations in Prob- lems I through 30. Primes denote derivatives with respect to x throughout. 17. y = (4x + y)? 19. x'y' + 2xy = 5y 21. y' = y+y 23. xy' + 6y = 3xy/ 1. (x+ y)y = x-y 3. xy = y +2xy 5. x(x+ y)y' = y(x - y) 6. (x+2y)y' = y 7. xy'y x+y 9. x'y' = xy + y 11. (x- y)y 2ry 12. xyy' = y +x4x² + y 13. xy = y+ Vx2 + y 14. yy +x = ? + y? 2. 2xyy' = x? + 2y² 4. (x - y)y =x + y %3! 8. x'y = xy +x'ea 10. xyy' = x + 3y? 1.6 Substitution Methods and Exact Equations 73 In Ruoblams l thuouglt 12 uarif aiuar diffauantial diff tiel
15. x(x + y)y' + y(3x + y) = 0 16. y = Vx+y+T 18. (x + y)y = 1 20. y'y +2xy = 6x 22. x'y +2xy = 5y 24. 2xy + y'e-2 = 2xy 25. y(xy + y)(1 +x*)2 =x 26. 3y?y' +y =e* 27. 3xy'y 3x + y 28. xe'y = 2(e +x'e) 29. (2x sin y cos y)y 4x + sin' y 30. (x +e')y' = xe -1 Find general solutions of the differential equations in Prob- lems I through 30. Primes denote derivatives with respect to x throughout. 17. y = (4x + y)? 19. x'y' + 2xy = 5y 21. y' = y+y 23. xy' + 6y = 3xy/ 1. (x+ y)y = x-y 3. xy = y +2xy 5. x(x+ y)y' = y(x - y) 6. (x+2y)y' = y 7. xy'y x+y 9. x'y' = xy + y 11. (x- y)y 2ry 12. xyy' = y +x4x² + y 13. xy = y+ Vx2 + y 14. yy +x = ? + y? 2. 2xyy' = x? + 2y² 4. (x - y)y =x + y %3! 8. x'y = xy +x'ea 10. xyy' = x + 3y? 1.6 Substitution Methods and Exact Equations 73 In Ruoblams l thuouglt 12 uarif aiuar diffauantial diff tiel
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
please send handwritten solution for Q 4 1.6
Transcribed Image Text:1.6 Problems
Find general solutions of the differential equations in Prob-
lems i through 30. Primes denote derivatives with respect to x
throughout.
15. x(x + y)y' + y(3x + y) = 0
16. y = Vx + y +1
18. (x + y)y = 1
20. y?y' + 2xy = 6x
22. x'y + 2xy = 5y*
24. 2xy' + y'e-24 = 2xy
25. y?(xy' + y)(1 +x*)/2 = x
26. 3y?y' + y' = e*
27. 3xyy = 3x* + y
28. xe' y = 2(e +x'e)
29. (2x sin y cos y)y' 4x +sin' y
30. (x +e)y' = xe – 1
17. y = (4x + y)?
19. xy' + 2ry 5y
21. y' = y + y
23. xy' + 6y = 3xyt/3
1. (x + y)y = x - y
3. xy = y +2xy
5. x(x+ y)y' = y(x – y) 6. (x + 2y)y' y
7. xy'y' = x + y
9. x'y = xy + y
11. (x² – y?)y = 2ry
12. xyy' = y? + x/4x? + y?
13. xy = y +Vr + y?
14. yy +x = x² + y?
2. 2xyy' = x? + 2y?
4. (x - y)y' = x+ y
%3D
8. x'y' = xy + x'erla
10. xyy' = x? +3y?
1.6 Substitution Methods and Exact Equations 73
In Problems 31 through 42, verify that the given differential
equation is exact; then solve it.
60. Use the method in Problem 59 to solve the differential
equation
dy
2y-x +7
31. (2x + 3y) dx + (3x +2y) dy = 0
32. (4x - y) dx + (6y – x) dy = 0
33. (3x? +2y?) dx+ (4xy +6y?) dy = 0
34. (2xy + 3x?) dx + (2x'y+4y) dy = 0
35. (x' +2) dx + (y² + In x) dy = 0
dx 4x – 3y – 18
61. Make an appropriate substitution to find a solution of the
equation dy/dx = sin(x – y). Does this general solution
contain the linear solution y(x) = x - 1/2 that is readily
verified by substitution in the differential equation?
62. Show that the solution curves of the differential equation
36. (1 + ye") dx + (2y + xe) dy = 0
y(2r - y)
x(2y3 - x)
37. (cos x + In y) dx +
dy = 0
dy
dx
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