Differential Pulse Code Modulation (DPCM)
Eq. $(5)$ in your question is really just a consequence of Eqs $(1)$, $(3)$, and $(4)$, so you have four independent equations. But that's all you need because there's a simple one-to-one correspondence between $\hat{d}[n]$ and $c[n]$ (given by the encoder). Note that the quantization error $e[n]$ is usually modeled as a zero-mean white random process with a uniform PDF.

You generally cannot compute an analytical solution of all the sequences in the coder due to the quantizer, which is of course a non-linear device. However, you can compute the sequences sample by sample by simply simulating the DPCM system as shown in the diagram in your question.

Assuming that the prediction filter has an initial state of zero (i.e., $\tilde{x}[0]=0$), and that the quantization step $\Delta=1$ for simplicity, we can easily compute all sequences given the input sequence. With an input sequence of x=[2,3,5,7] we obtain

$\begin{array}{cccrcr}
n &amp; x[n] &amp; \tilde{x}[n] &amp; d[n] &amp; \hat{d}[n] &amp; \hat{x}[n] \\
0 &amp; 2 &amp; 0.00 &amp; 2.00 &amp; 1 &amp; 1.00 \\
1 &amp; 3 &amp; 0.80 &amp; 2.20 &amp; 1 &amp; 1.80 \\
2 &amp; 5 &amp; 1.44 &amp; 3.56 &amp; 1 &amp; 2.44 \\
3 &amp; 7 &amp; 1.95 &amp; 5.05 &amp; 1 &amp; 2.95
\end{array}$