Find the basis of kernal and image of a transformation

Suppose that n is a positive integer, and let the transformation $\displaystyle T:C(x) \rightarrow C(x)$ be defined by $\displaystyle T(P(x))=xp'(x)-np(x) \ \ \ \ \ p(x) \in C(x) $ where C(x) is the set of continuous function mapping from $\displaystyle \mathbb {N} $ to a set M.

Find a basis of kernal(T) and Image(T)

so far:

So I know that we need to find the set $\displaystyle K = \{ p(x) \in C(x) : T(p(x)) = 0 \} $

Now, if $\displaystyle T(p(x)) = xp'(x)-np(x) = 0 $

we will have $\displaystyle p'(x)= \frac {np(x)}{x} $

So I need to find the set of function that have that behavior?

Re: Find the basis of kernal and image of a transformation

Quote:

Originally Posted by

**tttcomrader** Suppose that n is a positive integer, and let the transformation $\displaystyle T:C(x) \rightarrow C(x)$ be defined by $\displaystyle T(P(x))=xp'(x)-np(x) \ \ \ \ \ p(x) \in C(x) $ where C(x) is the set of continuous function mapping from $\displaystyle \mathbb {N} $ to a set M.

Find a basis of kernal(T) and Image(T)

so far:

So I know that we need to find the set $\displaystyle K = \{ p(x) \in C(x) : T(p(x)) = 0 \} $

Now, if $\displaystyle T(p(x)) = xp'(x)-np(x) = 0 $

we will have $\displaystyle p'(x)= \frac {np(x)}{x} $

So I need to find the set of function that have that behavior?

This is a separable ODE.

$\displaystyle \frac{dp}{dx}=\frac{np}{x} \iff \frac{dp}{p}=\frac{n}{x}$

Integrating both sides gives

$\displaystyle \ln|p|=n\ln|x|+c \iff p(x)=Ax^n$