Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Double checking needed on 2 Differential Equations

  1. Jun 7, 2005 #1
    Problem 1:

    Find solution for:

    [tex](x^2\,-\,4)\,y''\,+\,(3\,x)\,y'\,+\,y\,=\,0[/tex]

    using power series methods.

    Answer 1:

    I get a recursion formula:

    [tex]a_{n\,+\,2}\,=\,\frac{n\,+\,1}{4\,(n\,+\,2)}\,a_n[/tex]

    and a final answer:

    [tex]y(x)\,=\,a_0\,\left[1\,+\,\frac{x^2}{8}\,+\,\frac{3}{128}\,x^4\,+\,\frac{5}{1024}\,x^6\,+\,...\right]\,+\,a_1\,\left[x\,+\,\frac{x^3}{6}\,+\,\frac{x^5}{30}\,+\,\frac{x^7}{140}\,+\,...\right][/tex]

    Does that look right?



    Problem 2:

    Use Euler's method to solve:

    [tex](2\,x^2)\,y''\,+\,(x)\,y'\,+\,y\,=\,0[/tex]

    Answer 2:

    Using the quadratic equation to solve for r:

    [tex]2\,r^2\,-\,r\,+\,1\,=\,0[/tex]

    [tex]r\,=\,\frac{1}{4}\,\pm\,\frac{\sqrt{7}}{4}\,i[/tex]

    Which means that:

    [tex]\lambda\,=\,\frac{1}{4}[/tex] AND [tex]\mu\,=\,\frac{\sqrt{7}}{4}[/tex]

    And finally:

    [tex]y(x)\,=\,C_1\,x^{\frac{1}{4}}\,cos\,(\frac{\sqrt{7}}{4}\,ln\,x)\,+\,C_2\,x^{\frac{1}{4}}\,sin\,(\frac{\sqrt{7}}{4}\,ln\,x)[/tex]

    Thanks for the checking in advance!
     
    Last edited: Jun 7, 2005
  2. jcsd
  3. Jun 7, 2005 #2
    Mathematica agrees with both of your solutions. Well done.

    --J
     
  4. Jun 7, 2005 #3
    Thanks alot

    I need to get that program someday!
     
  5. Jun 7, 2005 #4
    It's possible that your college has a license for it and will give it to you. Why don't you contact your IT department and ask?

    --J
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook