Hermite differential equation: H_n(t) = (-1)^n e^(t^2) (d/dt)^n e^(-t^2)

[PhysicsCat]

Hello!

I have the following task:
"Show the following:
[math]H_n(t) = (-1)^ne^{t^2} \left(\frac{d}{dt} \right)^n e^{-t^2}[/math]i) is a polynomial of n-th degree.
ii) it satisfys the hermite differential equation: [math]x^{\prime \prime}(t) -2tx^{\prime}(t) + 2nx(t) = 0[/math] with [math]n \in \mathbb{N}[/math].
Notice: Use Induction."

Im not sure how to do any of those tasks, so Im thankful for every helping hand!

 

[Dr.Peterson]

Hello!

I have the following task:
"Show the following:
[math]H_n(t) = (-1)^ne^{t^2} \left(\frac{d}{dt} \right)^n e^{-t^2}[/math]i) is a polynomial of n-th degree.
ii) it satisfys the hermite differential equation: [math]x^{\prime \prime}(t) -2tx^{\prime}(t) + 2nx(t) = 0[/math] with [math]n \in \mathbb{N}[/math].
Notice: Use Induction."

Im not sure how to do any of those tasks, so Im thankful for every helping hand!

Why were you assigned this if you have no idea how to do any of it? Most likely you know something about ideas like differentiation and induction, but are unwilling to try anything without having a full understanding of the problem. That is not the way to solve problems.

Have you tried just evaluating it for n=1 (or, if it doesn't bother you, for n=0)? That should give you some sense of how it can make sense at all.

Now please show some sort of work, or at least tell us what you do know. If you try something and get stuck, then we'll know how to help.

 

[PhysicsCat]

Hello!

You are right, I am sorry.

My ideas:
i)
For n=0 and n=1 we get: [imath]H_0(t) = 1[/imath] and [imath]H_1(t) = 2t[/imath]. These are obviously polynomials.
If we now assume, that [imath]H_n(t)[/imath] is also a polynomial, we now have to show that [imath]H_{n+1}(t)[/imath] is a polynomial.
[math]H_{n+1} = \left( -1 \right)^{n+1} e^{t^2} \left( \frac{d}{dt} \right)^{n+1}e^{-t^2} = \left( -1 \right) \left(-1 \right)^n e^{t^2} \left( \frac{d}{dt} \right)^n(-2t) \cdot e^{-t^2}[/math]This is, where I get stuck and dont know how to do the induction step.

ii)
We know from i), that [imath]H_0(t) = 1[/imath] and [imath]H_1(t) = 2t[/imath]. Plug them in the diff equation seperately and we get for [imath]H_0(t)[/imath]:
[math]H_0^{\prime \prime}(t) -2tH_0^{\prime} + 2nH_0(t) = 0[/math][math]\Longleftrightarrow 0 -2t \cdot 0 +2 \cdot 0 \cdot 1 = 0[/math]Which is true. For [imath]H_1(t)[/imath], we get:
[math]H_1^{\prime \prime}(t) -2tH_1^{\prime} + 2nH_1(t) = 0[/math][math]\Longleftrightarrow 0 - 2 \cdot 2t + 2 \cdot 1 \cdot 2t = 0[/math]Which is also true. Lets now assume, that [imath]H_n(t)[/imath] does solve the hermite equation. We need to show, that [imath]H_{n+1}[/imath] also solves it.
[math]H_{n+1}^{\prime \prime}(t) -2tH_{n+1}^{\prime} + 2nH_{n+1}(t) = 0[/math]This is where I get stuck on this section. I dont know how to express [imath]H_{n+1}[/imath] in magnitudes of [imath]H_n[/imath], in order to do the induction step.

Thank you for your help and have a nice day!