= Solution
Take the <Fourier transform> in $x$. The transform of the source is $\delta(t-t')e^{-ikx'}$, so the <Green function> equation becomes
$$
\partial_t\widetilde G+Dk^2\widetilde G=\delta(t-t')e^{-ikx'}.
$$
Choose the <causal Green function>, zero for $t<t'$. Away from $t'$ the solution is a decaying exponential, and integrating across $t'$ gives the jump $\widetilde G(t'+)-\widetilde G(t'-)=e^{-ikx'}$. Therefore
$$
\boxed{\widetilde G=H(t-t')e^{-ikx'}e^{-Dk^2(t-t')}.}
$$
The phase shifts the kernel's centre to $x'$. Applying the inverse <Fourier transform> and the <Fourier transform of a Gaussian> with $a=D(t-t')$ gives
$$
\boxed{G=\frac{H(t-t')}{\sqrt{4\pi D(t-t')}}\exp\left[-\frac{(x-x')^2}{4D(t-t')}\right].}
$$
For positive elapsed time this <heat kernel> solves the homogeneous diffusion equation and has unit integral; as elapsed time tends to zero it approaches the spatial delta distribution, supplying the required impulse.
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