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# A6 - Applied Math 250 Assignment#6 Spring 2009 due June...

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Unformatted text preview: Applied Math 250 Assignment #6 Spring 2009 due June 23rd A/ Problem Set 3 #5, #10, #11, #12, #13, B/ On Assignment #5 you saw that while linear, constant—coefﬁcient equations are the most well—understood, there are some other types of second—order equations that we can still solve (those in which either the dependent or independent variable does not appear explicitly). There are a few others; here’s an important one (these arise in more advanced applications as a consequence of switching from Cartesian to polar coordinates): An equation of the form 3323/” + aary’ + by = F (x) is called a “Cauchy-Euler equation” (or, sometimes, an “equidimensional equation”). These can be solved in a similar fashion to the constant-coefﬁcient case, except that we assume solutions of the form y = mm (instead of y = 6"”). i) Show that this assumption leads to the requirement that m2 + (a — 1)m + b = 0 (you may assume that a: > 0). As in the constant coefﬁcient case, this ” characteristic equation” may have 2 real, 1 real (repeated), or 2 complex (conjugate) roots, so we need three cases: 1. Distinct Real Roots: y = 0132"” + 0233’”. 2. Complex Roots: y = Clma+w +0230"m = :r°‘(Clx‘iB+C'2:r‘iﬂ). By writing mm as em” = em” = cos()61nx) + isinw in x) (and similarly for Cit—m), we can write the general solution to this case as y = 1:“ [A 003w ln :r) + B sin(ﬁ ln\$)] 3. Repeated Real Roots: This case is harder to explain. In the constant- coefﬁcient case, we found that multiplication by 1 generated our second (lin- early independent) solution; in this case it turns out that the required factor is In 23; the general solution is y = (Cl + 021H\$)\$m. ii) Solve the equation 3:22;” + my’ — 4y : 0. Assume a: > 0. iii) Solve the equation 2723/" + 7133/ + 9y = 0. Assume x > 0. iv) Solve the equation x23)” — asy’ + 5y = 0. Again, assume :5 > 0. As an exercise, you may wish to verify that your solutions do in fact satisfy the given equations. ...
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