p02 - F ( x, y ) d x + G ( x, y ) d y = 0. Is this equation...

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MATH201: Problem set 2 Set 2009/10/05 ; Due 2009/10/12 10:00 a.m. For each of the di±erential equations below, answer the questions: (a) Can this di±erential equation be solved by the method of Separation of Variables? ie can you write the di±erential equation in the form d y d x = f ( x ) g ( y )? If you can, ²nd the solution using the method of Separation of Variables (b) Is this di±erential equation homogeneous, or can it be tranformed to the homogeneous type? ie can you write the di±erential 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, ²nd the solution by using the substitution y = xv ,resp . Y = XV . (c) Is this di±erential equation linear? ie can you write the di±erential equation in the form d y d x + P ( x ) y = Q ( x )? If you can, ²nd the integrating factor exp( R P ( x )d x ) and hence the solution. (d) Multiply through by d x
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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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