How to prove this question by induction

[hannahang]

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

 

[phosgene]

Do you know the basics of proof by mathematical induction? You haven't stated where exactly you're having trouble.

 

[hannahang]

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..

 

[phosgene]

You're on the right track, but you're missing a few steps. For the induction step, you assume that

$\frac{d^n}{dx^n}e^{x^2} = P_{n}(x)e^{x^2}$

is true. Now use this to prove that

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

is true.

Here's a hint, what is

$\frac{d^{(n+1)}}{dx^{(n+1)}}e^{x^2}$

equal to in terms of n instead of (n+1)?

 

[hannahang]

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

 

[phosgene]

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 $P_{k}(x)$. We only need it to be a polynomial whose leading term is $2^{k}x^{k}$. I'm unsure what your last step is, but try to simplify your second to last step. You should get $e^{x^2}$ multiplied by a polynomial. Does this polynomial satisfy our definition of $P_{k}(x)$?

 

[hannahang]

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

 

[hannahang]

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

 

[phosgene]

I don't have the time to write it out in detail, but what you should have at the moment is

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

Factorise the $e^{x^2}$ and then tell me the resulting polynomial that it's multiplied by, the definition of $P_{(k+1)}(x)$ and whether or not your polynomial satisfies this.

 

[Mark44]

Please show me how you would solve this problem so that I can compare with mine...

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

From the Rules (https://www.physicsforums.com/showthread.php?t=414380):

Under no circumstances should complete solutions be provided to a questioner, whether or not an attempt has been made.

Thxs so much...I need to finish this asap...

Thxs so much for ur help

 

[hannahang]

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