Table 1 Main operations of the DTM
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) | \({{\sum _{r=0}^{k}\sum _{l=0}^{r}}}U\left( l\right) V\left( r-l\right) W\left( k-r\right) \) |
\(\dfrac{d^{n}}{dt^{n}}[u(t) ]\) | \(\left( k+1\right) \ldots \left( k+n\right) U\left( k+n\right) \) |
\(e^{\lambda t} \) | \(\dfrac{\lambda ^{k}e^{\lambda t_{0}}}{k!}\) |
\(\sin \left( \omega t\right) \) | \(\dfrac{\omega ^{k}}{k!}\sin \left( \omega t_{0}+\dfrac{\pi k}{2}\right) \) |
\(\cos \left( \omega t\right) \) | \(\dfrac{\omega ^{k}}{k!}\cos \left( \omega t_{0}+\dfrac{\pi k}{2}\right) \) |