# Table 5 Differential transforms${\mathbit{G}}_{\mathbit{i}}\mathbf{\left(}\mathbit{k}\mathbf{\right)}$for$\mathbit{g}\mathbf{\left(}\mathbit{t}\mathbf{\right)}\mathbf{=}\mathbit{y}\mathbf{\left(}\mathbit{t}\mathbf{\right)}\mathbf{ln}\mathbf{\left[}\mathbit{y}\mathbf{\left(}\mathbit{t}\mathbf{\right)}\mathbf{\right]}$,$\mathbit{k}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{4}$
k ${\mathbit{G}}_{\mathbit{i}}\mathbf{\left(}\mathbit{k}\mathbf{\right)}$
0 ${Y}_{i}\left(0\right)ln\left[{Y}_{i}\left(0\right)\right]$
1 ${Y}_{i}\left(1\right)ln\left[{Y}_{i}\left(0\right)\right]+{Y}_{i}\left(1\right)$
2 ${Y}_{i}\left(2\right)ln\left[{Y}_{i}\left(0\right)\right]+{Y}_{i}\left(2\right)+\frac{{\left({Y}_{i}\left(1\right)\right)}^{2}}{2{Y}_{i}\left(0\right)}$
3 ${Y}_{i}\left(3\right)ln\left[{Y}_{i}\left(0\right)\right]+{Y}_{i}\left(3\right)+\frac{{Y}_{i}\left(1\right){Y}_{i}\left(2\right)}{{Y}_{i}\left(0\right)}-\frac{{\left({Y}_{i}\left(1\right)\right)}^{3}}{6{\left({Y}_{i}\left(0\right)\right)}^{2}}$
4 ${Y}_{i}\left(4\right)ln\left[{Y}_{i}\left(0\right)\right]+{Y}_{i}\left(4\right)+\frac{2{Y}_{i}\left(1\right){Y}_{i}\left(3\right)+{\left({Y}_{i}\left(2\right)\right)}^{2}}{2{Y}_{i}\left(0\right)}-\frac{{\left({Y}_{i}\left(0\right)\right)}^{2}{Y}_{i}\left(2\right)}{2{\left({Y}_{i}\left(0\right)\right)}^{2}}+\frac{{\left({Y}_{i}\left(1\right)\right)}^{4}}{12{\left({Y}_{i}\left(0\right)\right)}^{3}}$