The authors prove that if \(G\) is a differential algebraic group over a differentially closed field \(K\) of characteristic 0 the underlying set of which is \(K^n\), then there is a differential embedding of \(G\) into a unipotent algebraic group over \(K\). The proof uses a result of independent interest, which shows that an affine group scheme over any algebraically closed field of characteristic 0 whose coordinate ring is a polynomial ring in countably many indeterminates is pro-unipotent. The proof of this result proceeds by reducing to the case where the field is \(\mathbb{C}\) and using results from the homotopy theory of connected Lie groups. The authors also supply a model theoretic proof of the theorem of Lazard which states that an affine group scheme over an algebraically closed field of arbitrary characteristic the underlying variety of which is an affine \(n\)-space is a unipotent linear algebraic group.