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Math Help - Maximum number of linearly independent solutions

  1. #1
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    Maximum number of linearly independent solutions

    What is the maximum number of linearly independent solutions of the diff. eqn. d4y/dx4=0 with y(0)=1?
    And what is the maximum number when the condition given is y'(0)=0?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Maximum number of linearly independent solutions

    Quote Originally Posted by arindamnaskr View Post
    What is the maximum number of linearly independent solutions of the diff. eqn. d4y/dx4=0 with y(0)=1? And what is the maximum number when the condition given is y'(0)=0?
    Hint y(x) is a solution of y^{(4)}=0 if and only if y(x)\in\mathbb{R}_3[x] (i.e. a polynomial of degree \leq 3) .
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  3. #3
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    Re: Maximum number of linearly independent solutions

    More abstractly, an nth order linear differential equation will have n independent solutions (its solution space has dimension n) and adding one constraint will reduce that by 1.

    Less abstractly, it should be very easy to find the general solution just by integrating four times, then apply the conditions and see what you get!
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    Re: Maximum number of linearly independent solutions

    Quote Originally Posted by FernandoRevilla View Post
    Hint y(x) is a solution of y^{(4)}=0 if and only if y(x)\in\mathbb{R}_3[x] (i.e. a polynomial of degree \leq 3) .

    Thanks for the reply. I've got it.
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    Re: Maximum number of linearly independent solutions

    Thanks..
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