Differential Equations: Part I by L. W. F. Elen M.Sc. (auth.)

By L. W. F. Elen M.Sc. (auth.)

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Extra resources for Differential Equations: Part I

Example text

Cos x 4. dy y 3 2. dx =y . x + ~ = xy2. X ix - ~~ + Y y sin x + y2 = y4e X. = O. 27 ORDINARY FIRST-ORDER EQUATIONS 5. Xy2: - 2y 3 = 2x3 ; given that y = 1 when x = 1. ) 6. X: 7. + y = y2x 2 In x. ~~ + y 8. 2x: 9. ) = y3 see6 x. ) 1)y3; given that when x = 1, Y x = 1. given that y has a minimum value when x = t and show that this minimum value is approximately -1,44. ) ANSWERS 1. 3. ~= Y ! y 1 C 2. 2 = 2 y x Cx - x 2 • 1 4. "3 = tex = sin x + C eos x. tx. + Ce 3x . y 1 6. - = - x 2 In x Y 1 7. 2 = -2tanxsee2 x - ttan 3 xsee 2 x y 1 8.

27 ORDINARY FIRST-ORDER EQUATIONS 5. Xy2: - 2y 3 = 2x3 ; given that y = 1 when x = 1. ) 6. X: 7. + y = y2x 2 In x. ~~ + y 8. 2x: 9. ) = y3 see6 x. ) 1)y3; given that when x = 1, Y x = 1. given that y has a minimum value when x = t and show that this minimum value is approximately -1,44. ) ANSWERS 1. 3. ~= Y ! y 1 C 2. 2 = 2 y x Cx - x 2 • 1 4. "3 = tex = sin x + C eos x. tx. + Ce 3x . y 1 6. - = - x 2 In x Y 1 7. 2 = -2tanxsee2 x - ttan 3 xsee 2 x y 1 8. 7 - + Csee 2 x. 1 9. - = In x x3 . + x 2 + Cx.

SINGULAR SOLUTIONS This equation is of the type y = px + j(P) (1) where p = dyjdx. Differentiate with respect to x then p = p + x dp + l' (P) dp dx dx Hence ~ {f'(P) + x} either dp = 0 dx or From (1) and (2) = 0 p=C I'(P) + x = 0 y = Cx (2) (3) + j(C) which is the compiete primative and it represents a family of straight lines. Equation (3) needs further consideration. It is known that if f(x, y, c) = 0 where cis a parameter and if we eliminate c between this equation and aj/ac = 0 we obtain the envelope of the family of curves.

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