Thus we must use variation of parameters to nab the

Info icon This preview shows pages 2–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Obviously the driving function here is NOT a UC function. Thus, we must use variation of parameters to nab the culprit. Corresponding Homogeneous: F.S. = {sin(2 x ), cos(2 x )}. If y p = v 1 cos(2 x ) + v 2 sin(2 x ), then v 1 and v 2 are solutions to the following system: Solving the system yields v 1 = -2 and v 2 = 2cot(2 x ). Thus, by integrating, we obtain Thus, a particular integral of the ODE above is ______________________________________________________________________ 4. (10 pts.) Set up the correct linear combination of undetermined coefficient functions you would use to find a particular integral, y p , of the O.D.E. First, the corresponding homogeneous equation is which has an auxiliary equation given by 0 = m 2 - 6 m + 10. Thus, m = 3 + i or m = 3 - i , and a fundamental set of solutions for the corresponding homogeneous equation is { exp(3 x )cos( x ), exp(3 x )sin( x ) }. Taking this into account, we may now write ______________________________________________________________________ Bonus Noise: If , then the first n derivatives are nonzero and given by Higher order derivatives are zero. If f is a solution to (*) where m > n , then by direct substitution (**) The left side of (**) is a nonzero polynomial of degree at most n if there is some a k nonzero for some k with 0 k n . Such a polynomial has at most n roots. So in this case, we would not have a function identity on any interval of the line. It follows that a k = 0 for every k with 0 k n . Consequently, the fundamental set of solutions includes the following:
Image of page 2

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
TEST2/MAP2302 Page 3 of 4 ______________________________________________________________________ 5. (10 pts.) The factored auxiliary equation of a certain homogeneous linear O.D.E. with real constant coefficients is as follows: Note: The plus sign in the factor ( m + ( π - i )) 2
Image of page 3
Image of page 4
This is the end of the preview. Sign up to access the rest of the 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