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Unformatted text preview: MATH 3103 Fall 2011 Differential Equations Homework Set 6 Some Solutions The following solutions are not necessarily complete, but should highlight the main points for most questions. If you have any further questions regarding the problems and solutions, feel free to discuss them on WebCT. Solutions marked by * are taken from the publisher’s solutions for the textbook by Boyce and DiPrima, 9th edition (Wiley). Section 2.7: * Section 2.7 # 20 : [The last line of (c) should read “ t k +1 = t k + h ”] (This question was optional; the solution is included since the corresponding problem for handing in, #21 below, was much simpler.) * Section 2.7 # 21 : Section 3.1: * Section 3.1 # 8 : † Section 3.1 # 10 : † Section 3.1 # 15 : † Section 3.1 # 17 : * Section 3.1 # 20 : † Section 3.1 # 21 : * Section 3.1 # 23 : † Section 3.1 # 28 : (b) The roots are real and with opposite signs whenever c < 0 (for a > 0), since then always b 2 4 ac > b 2 ≥ 0. (c) follows immediately from (a)... Another approach : If r 1 and r 2 are roots of the characteristic equation ar 2 + br + c = 0, then we must have ar 2 + br + c = a ( r r 1 )( r r 2 ) = a ( r 2 ( r 1 + r 2 ) r + r 1 r 2 ) = 0 , so that the coefficients of the quadratic equation are related to the roots by r 1 + r 2 = b/a, r 1 r 2 = c/a....
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 Spring '11
 RWK
 Math, Differential Equations, Equations, Constant of integration, step size

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