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p02 - F x y d x G x y d y = 0 Is this equation exact Does...

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MATH201: Problem set 2 Set 2009/10/05 ; Due 2009/10/12 10:00 a.m. For each of the differential equations below, answer the questions: (a) Can this differential equation be solved by the method of Separation of Variables? ie can you write the differential equation in the form d y d x = f ( x ) g ( y )? If you can, find the solution using the method of Separation of Variables (b) Is this differential equation homogeneous, or can it be tranformed to the homogeneous type? ie can you write the differential equation in the form d y d x = f ( y/x ) or d Y d X = f ( Y/X ) for appropriately chosen X and Y ? If you can, find the solution by using the substitution y = xv , resp. Y = XV . (c) Is this differential equation linear? ie can you write the differential equation in the form d y d x + P ( x ) y = Q ( x )? If you can, find the integrating factor exp( P ( x ) d x ) and hence the solution. (d) Multiply through by d x in this the differential equation and write it in the form
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Unformatted text preview: F ( x, y ) d x + G ( x, y ) d y = 0. Is this equation exact? Does F/y = G/x ? If it is, nd the solution in the form W ( x, y ) = const. In all cases, even if the equation is not of a form appropriate for the method, try to make some obvious transformations which may bring it to such form. Please submit your answer to equation 10 for marking. NB: the full marks will only be awarded for all four methods, i.e. either solving equation 10 with that method or showing that the method is not applicable. 1. 1 y d y d x-1 x = 0 . 2. d y d x = 2-3 y 3. d y d x = x 3-y 3 4. xy d y d x = x 2 + y 2 5. d y d x = 3 x + 4 y + 5 4 x + 3 y + 7 6. y 3 d y d x + xy 2 = x 2 y 2 7. x 3 d y d x + y 3 = xy 8. ( x + 3 y 2 ) d y d x + y =-3 x 2 9. x d y d x + y-2 x = 0 . 10. * x 3 d y d x-x 2 = x 2 y *...
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