I'm having an issue. I have to solve a system of two equations, but the system is strange and none of my tools are being helpful. The issue is that the unknowns in the system are two functions.
The general idea here is we have
a(x)f(x) + b(x)g(x) = 1 c(x)f(x) + d(x)g(x) = 0
I have a(x), b(x), c(x), and d(x) given and need to find f and g.
What tools might be appropriate to solve this?
Thanks!
Same as if they were "normal algebraic " variables, i.e. $$\left\{ \begin{gathered} f(x) = \frac{{d(x)}} {{a(x)d(x) - b(x)c(x)}} \hfill \\ g(x) = - \frac{{c(x)}} {{a(x)d(x) - b(x)c(x)}} \hfill \\ \end{gathered} \right. $$ Of course you are going to have a ratio of functions, which means that you have to take care of the points where the denominator $(a(x)d(x) - b(x)c(x))$ becomes null (if any), unless they cancel with the zeros of the numerator, etc.