item number 22 44Equations of Order One|Ch. 233. (1 + t) ds + 2t[st² - 3(1 +t)] dt = 0; when t = 0, s = 2.%3D%3DAns. s= (1 + t)[3 – exp (-1]).Miscellaneous ExercisesIn each exercise, find a set of solutions, unless the statement of the exercise stipulatesotherwise.Ans. 2e = e2* + c.Ans. 2y = x* + cx?d. y = exp (2x- y)2. (x* + 2y) dx – x dy = 0.3. (3xy + 3y – 4) dx + (x + 1) dy = 0.4. (x + y) dx + x dy = 0.5. y dx – x(2x + 3y) dy = 0.6. (x2 + 1) dx +x'y dy = 0.:7. (x + y) dx + y'(3x + ky) dy = 0; k is constant.8. y =x' - 2xy; when x = 1, y = 2.9. sin 0 dr/do = -1- 2r cos 0.10. y(x + 3y) dx + x? dy = 0.11. dy/dx = sec? x sec' y.12 (2x – x?y +y) dx – x(2x' + y) dy = 0.(1+x)y = x*y*.14. y(3 + 2xy?) dx + 3(x?y? + x - 1) dy = 0.15. (2x? – 2xy – y?) dx + xy dy = 0.16. y(x? + y?) dx + x(3x² – 5y*) dy = 0; when x = 2, y = 1.Ans. y = 2(x + 1)- + (x + 1)-Ans. x(x + 2y) = c.%3DAns. y(x + y) = cx.Ans. xy = 3(1 + cx - x).Ans. ky + 4xy + x* = c.Ans. 2y = x? - 1+ 4 exp (1 - x²).%3D%3DAns. r sin? 0 = c + cos 0.Ans. xy = c(2x + 3y).!3!%3!Ans. 3 tan x +c = 3 sin y – sin-y.Ans. 2xy In cx| = 4x - y.Ans. x'y + 1 = y'(c + 3x - 3 arc tan xAns. x'y = 3(c + y - xy).Ans. x = cy - x) exp (y/x).%3D17. y + ay = b; a and b constants. Solve by two methods.18. (x – y) dx – (x + y) dy = 0. Solve by two methods.19. dx/dt = cos x cos? t.20. (sin y – y sin x) dx + (cos x +x cos y) dy = 0.21. (1 + 4xy – 4x?y) dx + (x - x) dy = 0; when x 2, y = .Ans. 2y – 2x'y + 3x = 0.Ans. y = b/a + ceAns. x - 2xy - y =Ans. 4 In sec x + tan x| = 21 + sin 2t + c.Ans. x sin y + y cos x = c.= c.%3D%3D%3!22 3x'y = 2yty- 3).23 (2y cos x + sin* x) dx = sin x dy; when x 7, y = 1.Ans. 2x*y = x? + 2x + 2 In (x – 1).Ans. y = cty – 3) exp (x-%3D%3D%3D24. xy(dx – dy) = x' dy + y? dx.25 a*(dy – dx) = x² dy + y? dx; a constant.Ans. y = 2 sin? x sin?Ans. x = y In |cxyl!26. (y – sin? x) dx + sin x dy = 0.27. (x + 2y) dx + (2x + y) dy = 0.28. (2xy – 3x?) dx + (x² + 2y) dy = 0.In solving Exercises 29 through 33, recall that the principal value arc sin x of theinverse sine function is restricted as follows: -n S arc sin x s tn.Ans. 2 arc tan (y/a) = In ic(x + a)/(x – a)\.Ans. y(csc x – cot x) = x + c – sin x.Ans. x? + 4xy + y? = c.Ans. x*y – x' + y² = c.%3D= C.25. I- y dx +V1-x² dy = 0.Ans. arc sin x + arc sin y = c, or a part of the ellipsex + 2c,xy + y? + c} – 1 = 0; where c, = cos c.

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44 Equations of Order One |Ch. 2 33. (1 + t) ds + 2t[st² - 3(1 +t)] dt = 0; when t = 0, s = 2. %3D %3D Ans. s= (1 + t)[3 – exp (-1]. Miscellaneous Exercises In each exercise, find a set of solutions, upless the statement of the exercise stipulates otherwise. Ans. 2e = e2* + c. Ans. 2y = x* + cx? d. y = exp (2x- y) 2. (x* + 2y) dx - x dy = 0. 3. (3xy + 3y - 4) dx + (x + 1) dy = 0. 4. (x + y) dx + x dy = 0. 5. y dx – x(2x + 3y) dy = 0. 6. (x2 + 1) dx +x'y dy = 0.: 7. (x + y) dx + y'(3x + ky) dy = 0; k is constant. 8. y =x' - 2xy; when x 1, y = 2. 9. sin 0 dr/d0 = -1- 2r cos 0. 10. y(x + 3y) dx + x? dy = 0. 11. dy/dx = sec? x sec' y. 12 (2x – x?y +y) dx - x(2x' + y) dy = 0. (1 + x)y = x*y*. 14. y(3 + 2xy?) dx + 3(x'y? + x- 1) dy 0. 15. (2x? – 2xy – y?) dx + xy dy = 0. 16. y(x? + y?) dx + x(3x² – 5y) dy = 0; when x = 2, y = 1. Ans. y = 2(x + 1)- + (x + 1)- Ans. x(x + 2y) = c. Ans. y(x + y) = cx. - x*). %3D Ans. xy = 3(1 + cx - Ans. ky* + 4xy' + x* = c. Ans. 2y = x? - 1 + 4 exp (1 - x2). Ans. r sin? 0 = c + cos 0. Ans. xy = c(2x + 3y). 3 sin y – siny. %3D %3D !3! %3D %3! %3D Ans. 3 tan x +c = Ans. 2xy In |cx| = 4x - y. Ans. x'y +1= y'(c + 3x - 3 arc tan x Ans. x'y = 3(c + y – xy). Ans. x = cdy - x) exp (y/x). 17. y + ay = b; a and b constants. Solve by two methods. 18. (x – y) dx – (x + y) dy = 0. Solve by two methods. 19. dx/dt = cos x cos? t. 20. (sin y – y sin x) dx + (cos x +x cos y) dy = 0. 21. (1 + 4xy – 4x?y) dx + (x - x) dy = 0; when x = 2, y = . Ans. 2y – 2x²y + 3x = 0. Ans. y = bja + ce Ans. x - 2xy - y = c. Ans. 4 In sec x + tan x = 21 + sin 2t + c. Ans. x sin y + y cos x = c. %3D %3D 22 3x'y = 2yty- 3). 23 (2y cos x + sin* x) dx = sin x dy; when x 7, y = 1. Ans. 2x*y = x? + 2x + 2 In (x - 1). Ans. y = cly - 3) exp (x %3D %3D %3! 24. xy(dx – dy) = x' dy + y? dx. 25 a*(dy – dx) = x² dy + y? dx; a constant. Ans. y = 2 sin? x sin? Ans. x = y In |cxyl! 26. (y – sin? x) dx + sin x dy = 0. 27. (x + 2y) dx + (2x + y) dy = 0. 28. (2xy – 3x?) dx + (x² + 2y) dy = 0. In solving Exercises 29 through 33, recall that the principal value arc sin x of the inverse sine function is restricted as follows: -n S arc sin x s tr. Ans. 2 arc tan (y/a) = In ic(x + a)/(x – a)\. Ans. y(csc x – cot x) = x +c - sin x. Ans. x? + 4xy + y? = c. Ans. x'y – x + y? = c. %3D 25. JI- y dx +1 - x² dy = 0. Ans. arc sin x + arc sin y = c, or a part of the ellipse x + 2c,xy + y? + c} - 1 = 0; where c, = cos c.