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    i am reviewing differential equations, and i wonder if you guys can clarify a point for me.

    In my book, for linear 2nd order o.d.e`s with constant coefficients, where the ode is:

    L(x)=\phi and \phi is the forcing function, and the solution to the associated homogeneous equation is:

    y_{h}=c_{1}+c_{2}e^{t}+c_{3}te^{  t}

    a couple of questions say: "determine the particular solution..."

    1) \phi = t book gives particular solution as:

    t*(at+b) but had previously said that:

    "if any term of the assumed solution - disregarding multiplicative constants - is also a term of the homogeneous solution, then the assumed solution must be modified...."

    i know that - but if you assume a soltion of the form at+b the

    b is a multiplicative constant - so the assumed solution should be correct, no? and why not?

    another one is where they have the forcing term as:

    2t^{2}-3t+82, y_{p}= t*(at^2+bt+c)

    in that case, is the thing about the multiplicative constant incorrect and they should not have said it?
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    (Original post by Hasufel)
    i am reviewing differential equations, and i wonder if you guys can clarify a point for me.

    In my book, for linear 2nd order o.d.e`s with constant coefficients, where the ode is:

    L(x)=\phi and \phi is the forcing function, and the solution to the associated homogeneous equation is:

    y_{h}=c_{1}+c_{2}e^{t}+c_{3}te^{  t}

    a couple of questions say: "determine the particular solution..."

    1) \phi = t book gives particular solution as:

    t*(at+b) but had previously said that:

    "if any term of the assumed solution - disregarding multiplicative constants - is also a term of the homogeneous solution, then the assumed solution must be modified...."

    i know that - but if you assume a soltion of the form at+b the

    b is a multiplicative constant - so the assumed solution should be correct, no? and why not?

    another one is where they have the forcing term as:

    2t^{2}-3t+82, y_{p}= t*(at^2+bt+c)

    in that case, is the thing about the multiplicative constant incorrect and they should not have said it?
    Isn't your particular solution at^2+bt and there is no t or t^2 term in the homogeneous solution so it's fine?

    Apologies if I misunderstood something you have said. The language of this is a distant memory for me.


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    (Original post by Hasufel)
    "if any term of the assumed solution - disregarding multiplicative constants - is also a term of the homogeneous solution, then the assumed solution must be modified...."

    i know that - but if you assume a soltion of the form at+b the

    b is a multiplicative constant - so the assumed solution should be correct, no? and why not?
    When it says disregarding multiplicative constants, it means that if f(t) is a term of the assumed solution, and A f(t) is a term of the homogeneous solution (where A is constant), then the two solutions do share a term - the multiplicative constant 'A' is not enough to say the solutions are different.

    So in this case, the homogeneous solution has the term c_1 and the assumed solution the term b, and they clearly differ only by a multiplicative constant (e.g. take A = c1/b), so the assumed solution does need modification.

    I think the confusion stems from the fact that you effectively have f(t) = 1 here. So you have a term c_1 f(t) which gets written as just c_1 and then it's easy to think "disregarding c_1 removes the term completely", which isn't what is actually meant.

    So I don't think they've said anything incorrect. What you've written isn't all that clear but to be honest I'm not sure how much is their fault and how much the editing you've done to post it here in a reasonably concise fashion.
 
 
 
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