Butterworth filter transfer function in time domain
A plot of the normalized impulse responses, for the n = 2 through 10 Butterworth low pass filters, are given by H.J. Blinchikoff, A.I. Zverev, "Filtering in the Time and Frequency Domains", Wiley-Interscience, John Wiley &amp; Sons, NY, ©1976, p. 113. This is shown below. They do not give the h(t) expressions in the book, at least where I have looked thus far. They say the figure is from Zverev's earlier book: A.I. Zverev, "Handbook of Filter Synthesis", Wiley, NY, ©1967.



EDIT: Looking back in my old computation notebooks, I found the impulse expression for the 3rd order Butterworth low pass filter. This pic is from 1983:



Hopefully, this is algebraically the same as your result.

EDIT 2: Gregg 1 gives $h(t)$ for the $M = 2$ Butterworth LPF:
$$h(t) = e^{-x}sin(x)U(t)$$
where $x = t/τ√2$, $τ = RC = 1/(2πB)$ and $B$ is the corner frequency in Hz. Butterworth LPFs are fine for many purposes, but not particularly good for others, e.g., they have relatively poor 1% settling times with step inputs 2.

References:


W.D. Gregg, “Analog and Digital Communication”, Wiley, NY, ©1977.
E. Voigtman, J.D. Winefordner, “Low-pass filters for signal averaging”, Rev. Sci. Instrum. 57 (1986) 957-966.


EDIT 3: Even though Matt has provided an excellent answer, for completeness I decided to test my 1983 $h(t)$ expression (see above) by comparing it with the numerically generated impulse response from a computer simulation $H(s)$ transfer function block. As per the OP, I assumed $2πB = 1$. The next figure shows that the results are the same.