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Math Help - Help with third order, non-linear DE.

  1. #1
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    Help with third order, non-linear DE.

    consider the thrid order DE:

    y'y'''=y''

    Question: Find all polynomials y(x) of degree 5 orl ess that satisfy this equation(It turns out that these are all of the solutions that are polynomials).

    How would I go about doing this using maple? Thanks in advance for any help.
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  2. #2
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    Looks to me by inspection, the only polynomial that satisfies the equation is P(x)=a+bx.
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  3. #3
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    I've done this in Mathematica before. It is more of an algorithm than a program, though. Let
    y(x) = ax^5 + bx^4 + cx^3 + dx + e

    Then calculate
    y'y''' - y''
    and set it equal to zero. Then you get a set of equations in a, b, c, d, and e by setting coefficients of powers of x equal to zero. In general you will get a system of equations to solve, which Maple can solve for you.

    However, in this case inspection works a lot faster.

    -Dan
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  4. #4
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    This is what I'd do in Mathematica: Define P_5(x). Calculate the first, third, and second derivatives, form the right and left sides, equate coefficients and construct seven equations in six unknowns (y'y''' has x^6 term), then use Solve to find the solution. Note that Solve reports c, d, e, and f are zero. Not sure why it's posting the results multiple times. Hopefully, you can translate it to Maple if you wish.

    Code:
    In[25]:= val = a + b*x + c*x^2 + d*x^3 + e*x^4 + f*x^5;
    d1 = D[val, {x, 1}];
    d3 = D[val, {x, 3}];
    d2 = D[val, {x, 2}];
    leftside = d1*d3;
    rightside = d2;
    eqns = Table[
      Coefficient[leftside, x, n] == Coefficient[rightside, x, n], {n, 0, 
       6}]
    Solve[eqns, {a, b, c, d, e, f}]
    
    
    Out[31]= {6 b d == 2 c, 12 c d + 24 b e == 6 d, 
     18 d^2 + 48 c e + 60 b f == 12 e, 96 d e + 120 c f == 20 f, 
     96 e^2 + 210 d f == 0, 360 e f == 0, 300 f^2 == 0}
    
    During evaluation of In[25]:= Solve::svars: Equations may not give \
    solutions for all "solve" variables. >>
    
    Out[32]= {{c -> 0, f -> 0, e -> 0, d -> 0}, {c -> 0, f -> 0, e -> 0, 
      d -> 0}, {c -> 0, f -> 0, e -> 0, d -> 0}, {c -> 0, f -> 0, e -> 0, 
      d -> 0}}
    [edit] results in 7 equations in 6 unknowns (y'y''' has x^6 term)
    Last edited by shawsend; August 9th 2008 at 06:38 PM. Reason: correct Mathematica code
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