MATH 255 Fall 2014 Assignment 5 Solutions - MATH 215/255...

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MATH 215/255 Fall 2014 Assignment 5 § 2.5, § 2.6 Solutions to selected exercises can be found in [Lebl], starting from page 303. 2.5.7: a) Find a particular solution of y 00 - 2 y 0 + y = e x using the method of variation of parameters. b) Find a particular solution using the method of undetermined coefficients.
The second equation is satisfied for u 2 = x , and the first then becomes u 0 1 = - x , which is satisfied for u 1 = - x 2 / 2. Our particular solution is now y p = - 1 2 x 2 e x + x 2 e x = 1 2 x 2 e x . Recalling the complementary solution (1), we have found the general form of the solution: y = C 1 e x + C 2 xe - x + 1 2 x 2 e x . b) The right-hand side is an exponential e x , so we try first y p = Ae x . This fails quite dramatically however, since we always obtain y 00 p - 2 y 0 p + y p = 0. A similar experiment with y = Axe x would yield the same result. We try finally y p = Ax 2 e x , for which y 0 p = ( Ax 2 + 2 Ax ) e x , y 00 p = ( Ax 2 + 4 Ax + 2 A ) e x . Then y 00 p - 2 y 0 p + y p = h ( Ax 2 + 4 Ax + 2 A ) - 2( Ax 2 + 2 Ax ) + Ax 2 i e x = 2 Ae x . We take therefore A = 1 / 2, so that after adding the complementary part (1), we find the same general solution as before: y = C 1 e x + C 2 xe - x + 1 2 x 2 e x . 2.5.9 For an arbitrary constant c find a particular solution to y 00 - y = e cx . Make sure to handle every possible real c .

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