MISCELLANEOUS TYPES OF DIFFERENTIAL EQUATIONS (a) Find the general solution of dy/dx 8x3 - 4x +1. (6) Determine the particular solution such that y(1) = 3. 2.58. 2.59. Solve each of the following...

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Extracted text: MISCELLANEOUS TYPES OF DIFFERENTIAL EQUATIONS (a) Find the general solution of dy/dx 8x3 - 4x +1. (6) Determine the particular solution such that y(1) = 3. 2.58. 2.59. Solve each of the following boundary-value problems. (a) y" = 28 Væ; v(1) = 0, y'(0) = -2 (b) xy" = 1+ t; y(1) = y'(1) = y"(1) = 0 (c) d?s/dt2 + 12t = 16 sin t; 8 = 2, ds/dt = -4 at t = 0 dy da (e) (1- a2)y' = 4y; y(0) 1. +21 = 10; 1(0) = 0, (d) dy = *V1-y2 V1-2 2.60. Solve (a) = -2xy, (b) xy' + 3y = 0; v(1) = 2, (c) %3D 2.61. Solve (a) (x+ 2y) da + (2x – 5y) dy = 0 (c) (yet - e-v) dæ + (xe-y+ e) dy = 0 dy (b) dx 4 - 2x cos y - 2y3 sec2 2x 3y2 tan 2x - 2 sin y 3 4xy2 ; y(1) = -1 dy (d) dx 4xy + 6y2 2.62. Each of the following differential equations has an integrating factor depending on only one variable. Find the integrating factor and solve the equation. (a) (4y - a2) dx + x dy = 0 (b) (2xy2 – y) da + (2x – a?y) dy = 0 (c) 2 da + (2x - 3y – 3) dy = 0; y(2) = 0 (d) (2y sin a + 3y4 sin x cos x) da - (4y3 cos2 x + cos x) dy = 0 2.63. Solve each of the following differential equations given that each has an integrating factor of the form xPya: (a) (3y - 2xy8) dæ + (4x – 3x2y?) dy = 0 (b) (2xys + 2y) dæ + (x2y2 + 2x) dy = 0 2.64. Solve (a) (2x2 + 2y2-y) da + (a2y + y3 + a) dy = 0 (b) (2 + x - y2) dæ - y dy = 0 (a) + dy dx 2.65. Solve = 4x2; y(1) = 2, (b) xy' - 4y = x, (c) = 3(y+ 2x) + 1; y(0) = 0, (d) + 2y cot x = csc x. de dy Solve (a) dx 24 - , (b) a d = #2 + 3xy + v?, (0) dy 2x -3y 2.66. (d) (* - y)u' + 3y-5x 0. x2 de 4y + 3x dy de + y = x3y2, 2.67. Solve (а) ӕ (b) 2x2y' = 0y + y3. x + 2y - 4 y - 2x + 3' dy (b) * - y +1 2.68. Solve (a) %3D %3D de * - y -1 2.69. Solve dy/dx = x2 + 2xy + y2 + 2x + 2y, y(0) = 0. 2.70. Solve (a) y'2 + (y - 1)y' -v = 0, (b) (ay' + y)2 -3.