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 “CauchyEuler equation” (or, sometimes, an “equidimensional equation”). These can be solved in a similar
fashion to the constantcoefﬁcient 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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 Spring '08
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