All Questions
4
questions
0
votes
2
answers
78
views
Find a function $f$ such that $\int_0^{P(x)} f(t) dt = 1- e^{2P(x)}$
I'm trying to solve the following homework problem. It states as follows:
"Let $P(x)$ be a polynomial such that $P'(x) \neq 0$ for all values of $x$. Does there exist a continuous function $f$ such ...
- 7,273
9
votes
4
answers
259
views
"Natural" proof of $P\left(\frac{d}{dx}\right)\bigl(e^{xy}Q(x)\bigr)=Q\left(\frac{d}{dy}\right)\bigl(e^{xy}P(y)\bigr)$.
In the context of linear differential equations, I've stumbled upon the following identity for an arbitrary pair of polynomials $P$ and $Q$ with real or complex coefficients:
$$
P\left(\frac{d}{dx}\...
- 2,210
0
votes
1
answer
73
views
Proving a Hermite polynomial equality
For $$H_{k}(x)=\frac{(-1)^{k}}{\sqrt{k!}}\exp\left\{\frac{x^{2}}{2}\right\}\frac{d^{k}}{dx^{k}}\exp\left\{-\frac{x^{2}}{2}\right\}$$
I want to prove $H'_{k}(x)=\sqrt{k}H_{k-1}(x)$.
So far I have $$\...
- 1,038
3
votes
1
answer
82
views
Why is the solution to $y' = y^n$ always in polynomial form EXCEPT when $n = 1$?
Could someone explain (intuition-wise) why the differential equation
$$y' = y^n$$
for $n \in \mathbb{N}$ seems to always some kind of polynomial solution (or a ratio of polynomials, etc.) except ...
- 13.9k