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Homework Help: How to prove this question by induction

  1. May 15, 2013 #1
    Prove by induction that for all n≥ 1,

    dn/ dxn (e ^(x2) = Pn (x) e ^(x2)

    where Pn(x) is a polynomial in x of degree n with coefficient of x^n equal to 2^n

    I have problems trying to prove this question by mathematical induction. Please help...Really much appreciated
  2. jcsd
  3. May 15, 2013 #2
    Do you know the basics of proof by mathematical induction? You haven't stated where exactly you're having trouble.
  4. May 15, 2013 #3
    I have started the prove with n = 1, that part was okay, I managed to prove that it is true for n = 1.

    The part I am having trouble with is to prove n = k+1.

    For that part, I tried to use product rule to differentiate d/dx(d^k/dx^k(e^x2)

    I got 2^(k+1)x^(k+1) e^x2 + e^x2(k2^kx^k-1)

    This is the part I got stuck with...Please advise thxs..
  5. May 15, 2013 #4
    You're on the right track, but you're missing a few steps. For the induction step, you assume that

    [itex]\frac{d^n}{dx^n}e^{x^2} = P_{n}(x)e^{x^2}[/itex]

    is true. Now use this to prove that

    [itex]\frac{d^k}{dx^k}e^{x^2} = P_{k}(x)e^{x^2}[/itex], where k=n+1

    is true.

    Here's a hint, what is


    equal to in terms of n instead of (n+1)?
  6. May 15, 2013 #5

    Thxs for the prompt reply...

    I have done this d^(k+1)/dx^(K+1) (e^(x^2) = d/dx(d^k/dx^k (e^x^2)
    = (Pk (x). e^(x^2)
    Then I used product rule for this part which ended up with

    = 2^(k+1)x^(k+1) e^x2 + e^x2(k. 2^k. x^k-1)
    = e^(x^2) P k+1 (x) + this part I am not sure

    It should just be equal to e^(x^2) P k+1 (x)... I don't know how to deal with the other part

  7. May 15, 2013 #6
    It seems that you're now using k to take the place of n. That's okay, just clarifying that so I don't confuse you with my reply.

    You're correct so far. But remember the definition of [itex]P_{k}(x)[/itex]. We only need it to be a polynomial whose leading term is [itex]2^{k}x^{k}[/itex]. I'm unsure what your last step is, but try to simplify your second to last step. You should get [itex]e^{x^2}[/itex] multiplied by a polynomial. Does this polynomial satisfy our definition of [itex]P_{k}(x)[/itex]?
    Last edited: May 15, 2013
  8. May 15, 2013 #7
    the second last step....that's the part I am not how I can simplyfy to get that polynomial Pk(x)

    I have tried many ways...but I can't seem to get the proving right...

    Could you please show me your workings so I can check with mine?

  9. May 15, 2013 #8
    Please show me how you would solve this problem so that I can compare with mine...Thxs so much.....I need to finish this asap....

    Thxs so much for ur help
  10. May 15, 2013 #9
    I don't have the time to write it out in detail, but what you should have at the moment is

    [itex]e^{x^2}(2^{(k+1)}x^{(k+1)} + Q_{k}(x)) + e^{x^2}(M_{(k-1)}(x))[/itex] where [itex]Q_{k}(x)[/itex] and [itex]M_{(k-1)}(x)[/itex] are polynomials of degree k.

    Factorise the [itex]e^{x^2}[/itex] and then tell me the resulting polynomial that it's multiplied by, the definition of [itex]P_{(k+1)}(x)[/itex] and whether or not your polynomial satisfies this.
  11. May 15, 2013 #10


    Staff: Mentor

    That's not how it works here at Physics Forums.

    From the Rules (https://www.physicsforums.com/showthread.php?t=414380):
  12. May 15, 2013 #11
    Thank you so much for your help. I will try to figure out how to do it again with your hints.

    Hopefully, I can solve it as soon as possible. Thank you
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