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Knowing which particular integral to use for ordinary differential equations watch

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    I'm sure this is a very simple thing in general but I don't seem to get it. Is there a trick to spotting it? It it just about knowing which one to use? There's too many occasions where, for example, I've used Ae^t instead of Ate^t or the other way round. And even now, I'm doing a cauchy-euler differential equation. The RHS to which is 1 - 3t so I looked up differential equations in my FP2 book and from that I got I should use, At + B as Yp(x) but it's beginning to look wrong to me (it may actually be correct but I can't tell) and I'm thinking maybe I'm meant to know something more advanced than FP2 to solve the question. I'm really not sure and the more I look at things, the more I'm getting confused. Somebody help.
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    Normally what's on the RHS of your equation (the "forcing term" as I'll call it) will be a trig function, an exponential function or a polynomial. Usually your particular integral will simply be something of the same form.

    So if the forcing term is \cos 4x then you put y_p = A\cos 4x + B\sin 4x. If the forcing term is e^{3x} then you put y_p = Ae^{3x}. If the forcing term is x^2+3 then you put y_p=Ax^2+Bx+C. The key things to note here are that for trig functions you need to include both sines and cosines, because of the way their derivatives work; and for polynomials you need to match the highest power of the polynomial (you don't need to go higher).

    This sometimes doesn't work when the complimentary function forms part of the forcing term. So for example if your forcing term was e^{2x} and your characteristic polynomial was \lambda^2 - 3\lambda + 2 = (\lambda - 1)(\lambda - 2), then your complimentary function would contain an e^{2x} term. In this case, you'd notice that you can't solve for the particular integral, so you have to multiply by x; i.e. put y_p=Axe^{2x}. Similarly, say you had \sin 2x in your forcing term and your characteristic polynomial was \lambda^2 + 4, then the same thing would happen.

    The very worst-case scenario is if your forcing term is, say, e^{2x} and your characteristic polynomial is \lambda^2 - 4\lambda + 4 = (\lambda - 2)^2. Then your complimentary function would be (Ax+B)e^{2x}, and so even trying y_p=Cxe^{2x} wouldn't work because part of the complimentary function contains it. In this case you need to go even further and put y_p = Cx^2e^{2x}.

    Hope this helps.
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    (Original post by nuodai)
    Normally what's on the RHS of your equation (the "forcing term" as I'll call it) will be a trig function, an exponential function or a polynomial. Usually your particular integral will simply be something of the same form.

    So if the forcing term is \cos 4x then you put y_p = A\cos 4x + B\sin 4x. If the forcing term is e^{3x} then you put y_p = Ae^{3x}. If the forcing term is x^2+3 then you put y_p=Ax^2+Bx+C. The key things to note here are that for trig functions you need to include both sines and cosines, because of the way their derivatives work; and for polynomials you need to match the highest power of the polynomial (you don't need to go higher).

    This sometimes doesn't work when the complimentary function forms part of the forcing term. So for example if your forcing term was e^{2x} and your characteristic polynomial was \lambda^2 - 3\lambda + 2 = (\lambda - 1)(\lambda - 2), then your complimentary function would contain an e^{2x} term. In this case, you'd notice that you can't solve for the particular integral, so you have to multiply by x; i.e. put y_p=Axe^{2x}. Similarly, say you had \sin 2x in your forcing term and your characteristic polynomial was \lambda^2 + 4, then the same thing would happen.

    The very worst-case scenario is if your forcing term is, say, e^{2x} and your characteristic polynomial is \lambda^2 - 4\lambda + 4 = (\lambda - 2)^2. Then your complimentary function would be (Ax+B)e^{2x}, and so even trying y_p=Cxe^{2x} wouldn't work because part of the complimentary function contains it. In this case you need to go even further and put y_p = Cx^2e^{2x}.

    Hope this helps.
    Thank you, that was reallly really helpful

    Just want to confirm, using my Yp(t) = 1 - 3t would not be correct if e^t is a substitution for x in the original differential equation then? (if I have understood correctly. if not, then :facepalm:) I'd have to use the quadratic version, right?
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    (Original post by Preeka)
    Thank you, that was reallly really helpful

    Just want to confirm, using my Yp(t) = 1 - 3t would not be correct if e^t is a substitution for x in the original differential equation then? (if I have understood correctly. if not, then :facepalm:) I'd have to use the quadratic version, right?
    Hrm? If the RHS of your equation is 1-3t then the correct substitution to make for the particular integral is At+B. I'm not sure what you're getting at with e^t being a substitution for x and so on.
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    (Original post by nuodai)
    Hrm? If the RHS of your equation is 1-3t then the correct substitution to make for the particular integral is At+B. I'm not sure what you're getting at with e^t being a substitution for x and so on.
    Basically, I was trying to solve a cauchy-euler differential equation using the substitution x = e^t so essentially solving in terms of t and then reverting back to x for the final general solution. So consequently my complimentary function has terms which involve exponentials which is why I thought I was messing up my particular part by just putting the particular integral as a linear polynomial. I'm not sure how much of that made sense but I hope it did.
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    (Original post by Preeka)
    Basically, I was trying to solve a cauchy-euler differential equation using the substitution x = e^t so essentially solving in terms of t and then reverting back to x for the final general solution. So consequently my complimentary function has terms which involve exponentials which is why I thought I was messing up my particular part by just putting the particular integral as a linear polynomial. I'm not sure how much of that made sense but I hope it did.
    Ah right, that makes sense. In that case, make the substitution so it becomes a normal linear differential equation and then forget the x terms ever existed; just look at it as a differential equation w.r.t. t and solve it that way. Then you can substitute back right at the end. I can't help you much more though, because I'm not sure what your precise equation is.
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    (Original post by nuodai)
    Ah right, that makes sense. In that case, make the substitution so it becomes a normal linear differential equation and then forget the x terms ever existed; just look at it as a differential equation w.r.t. t and solve it that way. Then you can substitute back right at the end. I can't help you much more though, because I'm not sure what your precise equation is.
    Alright, thanks I suppose I'll give it one more go and see how it figures out. If it still seems dodgy, I'll post the equation up to help give a clearer idea. Thanks for the help though
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    x^2 f''(x) + 5x f'(x) - 12y = 1 - 3 log x

    This is the equation. Is using Yp as At + B reasonable after doing the whole x = e^t substitution?
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    (Original post by Preeka)
    x^2 f''(x) + 5x f'(x) - 12y = 1 - 3 log x

    This is the equation. Is using Yp as At + B reasonable after doing the whole x = e^t substitution?
    Yes; putting x=e^t will give you \ddot{y} + 4 \dot{y} - 12y = 1-3t, so putting y_p=At+B will tell you what the particular integral is.
 
 
 
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