# How to solve this kind of second order equations with variable coefficients?

Let $$f(x)$$ be a function satisfying the functional equation $$c_1(x)f(x)^2 + c_2(x) f(x) + c_3(x) f(x-1) + c_4(x) f(x+1) = 0,$$ where $$c_1, \ldots, c_4$$ are known functions. What can be said about $$f$$?

• In general, nonlinear functional equations can't be solved in closed form. When they depend on arbitrary "known" functions, it's pretty much hopeless. Well, $f(x) = 0$ is a solution. – Robert Israel Jan 9 at 19:50
• I could understand that someone views this question as too broad, or utterly hopeless. But voting "Not research level" seems pretty ridiculous. May be whatever information is known on the functions $c_i$ should be added to the question. – Jan-Christoph Schlage-Puchta Jan 13 at 11:20