1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Using Matrix and Abel's formula to find second solution

  1. Apr 27, 2013 #1
    1. The problem statement, all variables and given/known data

    There is a Differential Equations question that I am stuck:

    Consider the equation: [itex](t^2)(y'')-2(t)(y')+(t^2+2)y=0[/itex], where t>0

    It is given that y1= t*cos(t) is a solution

    (a) There are two formulations for determining the Wronskian, (1) based on the determinant of a matrix which involved a pair of fundamental solutions and their derivatives, and (2) in terms of Abel's formula. Allowing the constant in Abel's formula to be equal to one, equate the two expressions for the Wronskian to derive a first order differential equation for the second solution y2.

    (b) Solve the first order equation for this second solution, y2. Note: [itex]d/dt (tan(t)) = 1/(cos^2(t))[/itex]

    2. Relevant equations

    Abel's theorem: Wronskian = [itex]c*exp(-\int p(t)\,dt) [/itex]for y'' + p(t)y' + q(t)y =0
    Matrix method: Wronskian = (y1)(y2') - (y1')(y2)

    3. The attempt at a solution

    For the Wronskian I got W = (t*cos(t))(y2') - (cos(t) - t*sin(t))(y2). If I did my derivatives right, that should be correct. For Abel's formula, I got W = e^(t^2). However, I'm not sure if it's possible to solve equation (t*cos(t))(y2') - (cos(t) - t*sin(t))(y2) = e^(t^2). I think I did something wrong, but not sure what. Can someone help?
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?