We are given the matrix equation:

$ABC^T D B A^T C = AB^T$

where all matrices are assumed to be $n \times n$ and invertible. We are tasked with solving for $D$.

Step-by-step Solution:

1. Multiply both sides by $(ABC^T)^{-1}$ on the left: $(ABC^T)^{-1} \cdot ABC^T D B A^T C = (ABC^T)^{-1} \cdot AB^T$ Since $(ABC^T)^{-1} \cdot ABC^T = I$ (the identity matrix), the equation simplifies to: $D B A^T C = (ABC^T)^{-1} \cdot AB^T$

2. Multiply both sides by $C^{-1}$ on the right: $D B A^T = (ABC^T)^{-1} \cdot AB^T \cdot C^{-1}$

3. Multiply both sides by $(B A^T)^{-1}$ on the right: $D = (ABC^T)^{-1} \cdot AB^T \cdot C^{-1} \cdot (B A^T)^{-1}$ Since $B A^T$ is invertible, this is valid.

Thus, the solution for $D$ is:

$D = (ABC^T)^{-1} \cdot AB^T \cdot C^{-1} \cdot (B A^T)^{-1}$

This expression represents $D$ in terms of the matrices $A$, $B$, and $C$.

Would you like further clarification or more details on any step? Here are some questions to deepen the understanding:

1. What happens if $A$, $B$, or $C$ are non-invertible?
2. How would the result change if we transpose the entire equation?
3. Can we simplify the expression for $D$ further under special conditions (e.g., symmetric matrices)?
4. What would the solution look like if $D$ were diagonal?
5. How does the presence of the transpose affect the inversion process?

Tip: Always check the invertibility of matrices before applying inverse operations, as non-invertible matrices can drastically change the nature of the solution.