I need to integrate an equation of the form
$f''(x) g_1+f'(x) g_2=a g_3$
where $a$ is a constant and $g_1,g_2,g_3$ are functions of $x$ whose values on the nodes of integration are known. The boundary conditions are $f(x=0)=0$, $f(x=1)=f_0$, $f_0$ is a constant. $g_1,g_2,g_3$ are arrays of data, a value for each node.
My question is: should I interpolate $g_1,g_2,g_3$ and then use the NDSolve command, or it is possible to employ directly a numerical tecnique?