1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    [itex]\frac{d^{(n+1)}}{dx^{(n+1)}}e^{x^2}[/itex]

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

    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

    Thxs
     
  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?

    Thxs
     
  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

    Mark44

    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
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: How to prove this question by induction
  1. Prove by Induction. (Replies: 15)

  2. Prove by induction (Replies: 3)

  3. Prove by induction (Replies: 5)

  4. Prove by induction (Replies: 5)

  5. Proving by induction (Replies: 5)

Loading...