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Thread: Finishing part of DE

  1. #1
    Registered Senior Member
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    Finishing part of DE

    Given DE is D^2(y)+5Dy+4y=cos 2x ........where D=>(dy/dx)

    We are to find the general solution.(I have done it already)
    Is the particular integral unique?If not,find another PI and show that the general solution is the same for all PI.

    I did the first part and got the general solution as

    y=C1 exp(-x)+C2 exp(-4x)+(1/10)cos 2x

    I cannot understand why the PI would be non-unique?Can anyone please check if I am correct in having that answer?

  2. #2
    man of no words temur's Avatar
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    1,330
    Think of the equation as Ly=f where L is a linear operator. To find a general solution you probably solved the homogeneous problem Ly=0 first then found a particular solution by trying it with a probable form. Let us write it formally: let h be a solution of Lh=0, and let p be some particular solutioin of Lp=f. Then you can see that p+h is also a particular solution: L(p+h)=Lp+Lh=f+0=f. A nice way of saying it is that the particular solution is not unique because L has a nontrivial kernel. You are writing the general solution for Ly=0 means you finding the kernel of L, and adding a particular solution to it means you are parallel shifting this kernel. The kernel of L is a linear space so the soluton set of the nonhomogeneous equation is an affine space, and this can be found by shifting the kernel of L by any vector (part.sol.) of this affine space. Perhaps you can see the situation clearly if you think L as a matrix and y,h,p,f etc as vectors.
    Last edited by temur; 05-30-07 at 06:28 PM.

  3. #3
    Registered Senior Member
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    491
    Oh!!!I was wrong to write that.Using method of undetermined co-efficients,I assumed the form of solution: a sin 2x+b cos 2x
    and got a=(1/10) and b=0.

    Thank you very much for the conceptual help.

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