Can anyone please explain to me step by step how to solve this question: Find the fundamental solution $g_k$ for the operators $ \frac{ d^k}{(dx)^k}$ on the real line such that $k=0,1,2,...$
I want this so I can understand how usually we can find the fundamental solution using distribution.
Let us solve the question for $k=2$. We write the differential equation
$$u''=\delta_0.$$ By a standard integration exercise we can say that $$u'=H(x) + c_0,$$ where $H(x)$ is the Heaviside function and $c_0$ is an arbitrary constant. We integrate once again (note that in the right-hand side we have usual piece-wise continuous functions, hence usual integration can be applied) to obtain $$u = xH(x) + c_0x + c_1,$$ where $c_1$ is another arbitrary constant.
In a general case for $k>2$ you might want to start with a polynomial $P$ of degree $k-1$ and try to find the solution of $u^{(k)}=\delta_0$ in the form $H(x)P(x)$. The polynomial is given by $k$ coefficients and you have essentially $k$ equations.