If y''(t)+p(t)y=0 is nonoscillatory then y''(t)+p(t)y=f(t) is nonoscillatory. p(t) =>0 and f(t)=>0 and f(t) continuous function.
You need just show that if the non-homogeneous problem has oscillatory solutions, then the homogeneous problem has oscillatory solutions.
Now, if y solves the non-homogeneous problem and w solves the homogeneous one, then y-w solves the non-homogeneous problem.