$$Ly=h(x)$$

f is the linear combination of all the functions that are L(function)=0, so that
$$Lf=0$$
And g is ONE OF the functions that do this:
$$Lg=h(x)$$
Why is the general (i.e. total and only (minus degrees of freedom for unknowable coefficients)- is this the correct definition?) solution for y equal to this:
$$y=f+g$$ I understand that it WORKS, but I've read in many places to stop here, as I've found the 'general' solution (whatever that really means) despite only using 1 out of potentially many g that do this $$L(function)=h(x)$$?

Does this rule only apply to a subset of differential equations, or are all other g not independent from out 1st g?

Question

$$Ly=h(x)$$

f is the linear combination of all the functions that are L(function)=0, so that
$$Lf=0$$
And g is ONE OF the functions that do this:
$$Lg=h(x)$$
Why is the general (i.e. total and only (minus degrees of freedom for unknowable coefficients)- is this the correct definition?) solution for y equal to this:
$$y=f+g$$ I understand that it WORKS, but I've read in many places to stop here, as I've found the 'general' solution (whatever that really means) despite only using 1 out of potentially many g that do this $$L(function)=h(x)$$?

Does this rule only apply to a subset of differential equations, or are all other g not independent from out 1st g?