I know from linear algebra that for a non-homogeneous consistent linear system that has an infinite

There is in fact a very strong connection between (certain kinds of) differential equations and systems of linear equations. They are both instances of linear transformations.
Consider the collection of all infinitely differentiable functions. There is a function $D$ from this set to itself that maps a function $f(x)$ to
$D(f)=f^{″}(x)(Ax+B{)}^2+f(x).$
This map is linear: D(f+g)=D(f)+D(g), and $D(\mathrm{\alpha <!--\; \alpha \; -->}f)=\mathrm{\alpha <!--\; \alpha \; -->}D(f)$ for all infinitely differentiable functions f and g, and all real numbers $\mathrm{\alpha <!--\; \alpha \; -->}$.
In particular, fix a function $k_0(x)$, and suppose that $D(f)=k_0(x)$ (that is, f is a particular solution to $y^{″}(x)(Ax+B{)}^2+y(x)=k_0(x)$). Then:
1. If g(x) is any solution to D(g)=0, that is, $g^{″}(x)(Ax+B{)}^2+g(x)=0$, then f(x)+g(x) is also a solution to $D(y)=k_0(x)$. This because $D(f+g)=D(f)+D(g)=k_0(x)+0=k_0(x)$.
2. If h is any other solution to $D(y)=k_0(x)$, then there exists a solution g to D(g)=0 such that h(x)=f(x)+g(x). That is, all solutions to $D(y)=k_0(x)$ are obtained by taking a single particular solution and addding solutions to D(y)=0.
To see this, notice that if $D(f)=D(h)=k_0(x)$, then $D(h-<!--\; -\; -->f)=D(h)-<!--\; -\; -->D(f)=k_0(x)+k_0(x)=0$. So h−f is a solution to D(y)=0. And h=f+(h−f), so h can be obtained by adding a solution to D(y)=0 to f.

In particular, the differential equation
$y^{″}(x)(Ax+B{)}^2+y(x)=\text{some cubic}$
that has at least one particular solution has exactly as many particular solutions as there are solutions to
$y^{″}(x)(Ax+B{)}^2+y(x)=0.$
So if the homogeneous equation has infinitely many (essentially distinct) solutions, and your original equation has at least one particular solution, then the original equation will in fact have infinitely many (essentially distinct) solutions as well.
This is in complete analogy to the case of a system of linear equations, which can be seen as given by the matrix equation $A\mathbf{x}=\mathbf{b}$, where $A$ is the matrix of coefficients of the system: given any particular solution ${\mathbf{x}}_0$, you get all solutions by taking $x_0+\mathbf{z}$, where $\mathbf{z}$ is any solution to $A\mathbf{z}=\mathbf{0}$.
This is all part of linear algebra. Both systems of linear equations and linear differential equations can be studied together, and many general results apply to both, so it's no surprise you are seeing a lot of similarities: in fact, there are a lot of similarities.