Using Nevanlinna’s theory the author studies the algebraic differential equation \[ \Omega(z,w)=\sum^ n_{k=0}A_ k(z)w^ k \tag{*} \] with meromorphic coefficients \(A_ k\) and with a differential polynomial \(\Omega\) of weight \(\Delta\). One of the results is: If \(n>\Delta\) and if \(A_ q\), \(\Delta\leq q\leq n-1\), is a dominant coefficient of exponential type, then equation \((*)\) does not possess a \(\nu\)-valued algebroid solution \(w\) with a small number of branch points. This is a generalization of a theorem of Gundersen and Laine on meromorphic solutions.