Explicit solutions for linear system of PDEs with constant coefficients

Question

I've been recently trying to to solve the following system of linear 1st order PDE's:

$f:\Omega^d\xrightarrow{}\mathbb{R},\quad A^{(k)}\in\mathbb{R}^{N\times N},\quad B^{(k)}\in\mathbb{R}^N$

$\dfrac{\partial{y(\vec{x})}}{\partial{x_k}} = A^{(k)} y(\vec{x}) + B^{(k)} f(\vec{x}) \quad y(\vec{x}_0)=\vec{0} \quad k\in\{1,...,d\}$

Where $N>1$, $\Omega$ is a compact simply connected subset of $\mathbb{R}^d$. This seems quite standard but haven't managed to find conditions for uniqueness & existence of the solution and/or an explicit form which is it what I'm looking or eventually.

Is there a common book/reference or an explicit solution for these type of PDEs?

Thanks you for your help!

Note: I came across 1 which states a condition for uniqueness & existence in the case of locally integrable coefficients - that the matrices $A^{(k)}$ commute, in that case the homogeneous solution is easy to achieve but it seems that condition is too restricting in my case.

1 - ON SYSTEMS OF FIRST ORDER LINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH $L^p$ COEFFICIENTS - S. MARDARE

Answer 1

ADDDED: Since equation (1) below is a first order linear ODE,it has an explicit solution (with an integral). It follows that the solution to the entire system of equations can be written explicitly as an iterated integral.

Following up on @WillieWong's comments: If $$ [A^{(k)},A^{(j)}]=0,\ A^{(k)}B^{(j)}f - A^{(j)}B^{(k)}f= 0,\ B^{(k)}\partial_jf - B^{(j)}\partial_kf = 0,$$ then the system is said to be in involution. In that case, given any value for $y(x_0)$, there is a unique solution to the system. This is not hard to prove using the following inductive step:

For simplicity, assume $x_0 = 0$. For each $k \ge 0$, assume you have a solution $y$ along the set $x^{k+1}=\cdots x^{d}=0$. Then for each $(x^1, \dots, x^k) \in \mathbb{R}^k$, the equation \begin{equation}\tag{1} \dfrac{\partial{y(\vec{x})}}{\partial{x_{k+1}}} = A^{(k+1)} y(\vec{x}) + B^{(k+1)} f(\vec{x}) \end{equation} is a constant coefficient linear ODE with respect to $x^{k+1}$ along the curve obtained to restricting to the line where $(x^1, \dots, x^k)$ are held constant and $x^{k+2}=\cdots=x^d=0$. Given $y=0$ when $x^{k+1}=0$, there is a unique solution. If you do this for each $(x^1,\dots,x^k) \in \mathbb{R}^k$, then you get a function $y$ on the set $x^{k+2}=\cdots=x^d=0$. By induction, you get a function on all of $\mathbb{R}^d$.

This is not obviously a solution since each equation is solved only along a subset. But what you do now is the consider the equation, for each $j > k$, $$ \partial_j\left(\partial_ky - A^{(k)}y - B^{(k)}f\right)=0 $$ By the construction of $y$ above, this equation is known to hold when $i$. You now use the assumptions on $A^{(k)}$, $B^{(k)}$, $f$ to show that this system is equivalent to $$ \partial_k\left(\partial_jy - A^{(j)}y - B^{(j)}f\right)=0. $$ An inductive argument now shows that the entire system holds on $\mathbb{R}^n$.

If the system is not in involution, I'm not sure what happens, but I suspect that no solution exists. This theory is usually presented using Cartan's formulation using exterior differential systems or the more modern approach using Spencer cohomology. As far as I know, the only explicit description of this theory applied to a constant coefficient linear first order system of PDEs is my monograph Involutive Hyperbolic Differential Systems. Unfortunately, when I took a look a few years ago, I found it unreadable.

ADDED: The description above of a linear first order involutive system of PDEs was first found by Guillemin and is known as Guillemin normal form. See https://rdcu.be/cN0pn