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Table 1 Main operations of the DTM

From: A new multi-step technique with differential transform method for analytical solution of some nonlinear variable delay differential equations

Function Differential transform
\(\alpha u(t)\pm \beta v(t) \) \(\alpha U( k) \pm \beta V( k) \)
u(t)v(t) \({{\sum _{r=0}^{k}}}U\left( r\right) V\left( k-r\right) \)
u(t)v(t) w(t) \({{\sum _{r=0}^{k}\sum _{l=0}^{r}}}U\left( l\right) V\left( r-l\right) W\left( k-r\right) \)
\(\dfrac{d^{n}}{dt^{n}}[u(t) ]\) \(\left( k+1\right) \ldots \left( k+n\right) U\left( k+n\right) \)
\(e^{\lambda t} \) \(\dfrac{\lambda ^{k}e^{\lambda t_{0}}}{k!}\)
\(\sin \left( \omega t\right) \) \(\dfrac{\omega ^{k}}{k!}\sin \left( \omega t_{0}+\dfrac{\pi k}{2}\right) \)
\(\cos \left( \omega t\right) \) \(\dfrac{\omega ^{k}}{k!}\cos \left( \omega t_{0}+\dfrac{\pi k}{2}\right) \)