{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern