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40. (e' sin y+ tan y) dx + (e' cos y + x sec? y) dy = 0

40. (e' sin y+ tan y) dx + (e' cos y + x sec? y) dy = 0

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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please send handwritten solution for Q 40 1.6
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 31. (2x + 3y) dx + (3x +2y) dy = 0 dy 2y – x +7 dx 4x- 3y- 18 32. (4x – y) dx + (6y – x) dy = 0 33. (3x? + 2y?) dx + (4xy +6y?) dy = 0 34. (2xy + 3x?) dx + (2xy+4y) dy = 0 35. (x' +2) dx + (? + In x) dy = 0 36. (1 + ye") dx + (2y + xe) dy = 0 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 - n2 that is readily verified by substitution in the differential equation? 62. Show that the solution curves of the differential equation 37. (cos x + In y) dx + dy = 0 dy y(2r-y) dx x(2y-x) x +y 38. (x+ tan- y) dx + dy = 0 1+ y? 39. (3x?y' + y*) dx + (3x'y + y +4xy') dy = 0 40. (e' sin y + tan y) dx + (e cos y + x sec? y) dy = 0 are of the form x' + y' 3Cxy. 63. The equation dy/dx = A(x)y + B(x)y+C(x) is called a Riccati equation. Suppose that one particular solution yı(x) of this equation is known. Show that the substitution 3y 2y x2 - -- 41. dx + dy = 0 y = y + 2x/2 - 3y/3 42. 3y/ – 2r/2 dx + 2x5/2 y2/3 3x/2 y5/3 dy =0 transforms the Riccati equation into the linear equation
Transcribed Image Text: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 31. (2x + 3y) dx + (3x +2y) dy = 0 dy 2y – x +7 dx 4x- 3y- 18 32. (4x – y) dx + (6y – x) dy = 0 33. (3x? + 2y?) dx + (4xy +6y?) dy = 0 34. (2xy + 3x?) dx + (2xy+4y) dy = 0 35. (x' +2) dx + (? + In x) dy = 0 36. (1 + ye") dx + (2y + xe) dy = 0 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 - n2 that is readily verified by substitution in the differential equation? 62. Show that the solution curves of the differential equation 37. (cos x + In y) dx + dy = 0 dy y(2r-y) dx x(2y-x) x +y 38. (x+ tan- y) dx + dy = 0 1+ y? 39. (3x?y' + y*) dx + (3x'y + y +4xy') dy = 0 40. (e' sin y + tan y) dx + (e cos y + x sec? y) dy = 0 are of the form x' + y' 3Cxy. 63. The equation dy/dx = A(x)y + B(x)y+C(x) is called a Riccati equation. Suppose that one particular solution yı(x) of this equation is known. Show that the substitution 3y 2y x2 - -- 41. dx + dy = 0 y = y + 2x/2 - 3y/3 42. 3y/ – 2r/2 dx + 2x5/2 y2/3 3x/2 y5/3 dy =0 transforms the Riccati equation into the linear equation
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