Table 1 Main operations of DTM
Function | Differential transform |
|---|---|
\(\alpha u(t)\pm \beta v(t)\) | \(\alpha U_{k}\pm \beta V_{k}\) |
u(t)v(t) | \({{\sum _{r=0}^{k}}}U_{r}V_{k-r}\) |
\(\dfrac{d^{n}}{dt^{n}}[u(t) ]\) | \(k\left( k-1\right) \ldots \left( k+1-n\right) U_{k}\), \(k\ge n\) |
\(e^{\lambda t}\) | \(\dfrac{\lambda ^{k}e^{\lambda t_{0}}}{k!}\) |
\(\sin \left( \omega t\right)\) | \(\dfrac{\omega ^{k}}{k!}\sin \left( \omega t_{0}+\dfrac{\pi k}{2}\right)\) |
\(\cos \left( \omega t\right)\) | \(\dfrac{\omega ^{k}}{k!}\cos \left( \omega t_{0}+\dfrac{\pi k}{2}\right)\) |