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

A system of 1st order nonlinear differential equations

  1. Jun 18, 2010 #1
    Hello,

    Can you give some suggestions to solve the following system of 1st order nonlinear differential equations?

    Thank you.

    [tex]

    \[
    \begin{array}{l}
    u'(t) = Au^2 (t) + B(t)u + C(t) \\
    u(t) = \left[ {\begin{array}{*{20}c}
    {x_1 (t)} \\
    {x_2 (t)} \\
    \end{array}} \right] \\
    A = \left[ {\begin{array}{*{20}c}
    {a_{11} } & {a_{12} } \\
    {a_{21} } & {a_{22} } \\
    \end{array}} \right] \\
    B(t) = \left[ {\begin{array}{*{20}c}
    {f_{11} (t)} & {f_{12} (t)} \\
    {f_{21} (t)} & {f_{22} (t)} \\
    \end{array}} \right] \\
    C(t) = \left[ {\begin{array}{*{20}c}
    {g_{11} (t)} & {g_{12} (t)} \\
    {g_{21} (t)} & {g_{22} (t)} \\
    \end{array}} \right] \\
    \end{array}
    \]

    [/tex]
     
  2. jcsd
  3. Jun 18, 2010 #2

    Mark44

    Staff: Mentor

    What does u2(t) mean? Is it u(t) [itex]\cdot[/itex] u(t)?

    Also, shouldn't the differential equation be
    [tex]u'(t) = Au^2 (t) + B(t)u(t) + C(t) [/tex]
    ?
     
  4. Jun 18, 2010 #3

    HallsofIvy

    User Avatar
    Science Advisor

    As Mark44 notes, the "u^2" doesn't make sense here. If it is the dot product, then multiplying it by a two by two matrix doesn't make sense. If it is the cross product, then Both Au^2 and Bu are 2 dimensional vectors but you cannot add that to C(t), a two by two matrix.

     
  5. Jun 18, 2010 #4

    EnumaElish

    User Avatar
    Science Advisor
    Homework Helper

  6. Jun 18, 2010 #5

    Mark44

    Staff: Mentor

    For some reason, HallsOfIvy's reply didn't render correctly. Here it is.
     
  7. Jun 23, 2010 #6
    Thanks for your help and sorry for unclear things.

    u^2 is a cross product. It means
    [tex]


    \[
    u^2 (t) = \left[ {\begin{array}{*{20}c}
    {x_1^2 (t)} \\
    {x_2^2 (t)} \\
    \end{array}} \right] \\

    \]
    [/tex]

    And C(t)
    [tex]
    C(t) = \left[ {\begin{array}{*{20}c}
    {g_{1} (t)} \\
    {g_{2} (t)} \\
    \end{array}} \right] \\

    [/tex]
     
  8. Jun 23, 2010 #7
    Thanks EnumaElish. Is an analytical solution impossible? If there is a method to obtain an analytical solution, can you suggest me?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook