What is the relationship between linear, non-homogeneous system of differential equations and linear, non-homogeneous system of equations?

Question

I'm a student studying DE and Linear Algebra, and I noticed that in both processes of solving system of DEs and system of equations, we do things like finding particular solution, and add them with the homogeneous solution. I can imagine it in Linear Algebra when the particular solution move you to the span of the homogeneous solution, but why do we do the same in DE? Thank you! :D

Answer 1

That's a good observation! In both cases, we are finding the solutions of $T(X)=C$, where $T$ is a linear transformation and $C$ is a constant. Here, "linear" means that $T(aX+bY)=aT(X)+bT(Y)$, where $a$ and $b$ can take any constant value (usually from $\mathbb R$ or $\mathbb C$ in your case, but depending on situations, it can be from any field).

For systems of linear equations, $X$ is a vector in $\mathbb R^n$ (or $\mathbb C^n$) and we have $T(X)=AX$, where $A$ is the coefficient matrix. For systems of DEs, it is best illustrated with an example. If we have $f'(x)-f(x)=2x$, then we can take $T(f)=f'-f$ ($f$ can be any differentiable function) and $C=x\mapsto2x$. You can verify that these are linear.

The method of particular solutions works for all equations of this form, since if $T(X_0)=C$, then $T(X)=C\Leftrightarrow T(X)-T(X_0)=0\Leftrightarrow T(X-X_0)=0$, where $0$ is the zero vector of the vector space formed by the objects of interest (the zero vector in the case of linear equations and the constant zero function in the case of linear DEs).