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    • Thread Starter

    Give the general solution of this DE:


    I am able to get the complementary equation but I can't seem to finish off the question.
    Because there's a 3xe^x, I have to consider a particular integral. From my 'intelligent guess' I made yp(x)=(Ax+B).e^x. But when I differentiate that and continue, my A=0 which doesn't seem right.


    So this could be because I have used the wrong particular integral yp(x). If this is the case, what would it be?
    • PS Helper

    PS Helper
    When this happens you have to multiply your original guess of particular integral by x; that is, you need to substitute y_p = (Ax+B)xe^x. The reason why this happened is because e^x forms part of the complimentary function, which you should have got to be Ce^x + De^{4x}. If this had happened and you'd just had, say, 3e^x on the RHS, then substituting y_p=Axe^x would have done the trick, but since the forcing term depends on xe^x too we need to increase the power yet further.

    You don't really need to know any of what I just said; basically if you try a particular integral which is in the right form and it doesn't work, multiply the lot by x and try again. [You were right to try (Ax+B)e^x to start with though.]
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Updated: April 6, 2011

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