find the inverse laplace transform 𝐿−1[𝑠^2/(𝑠+𝑎)^3]
You asked:
Find the inverse laplace transform of \(L - 1 \cdot \frac{{s}^{2}}{{\left( s + a \right)}^{3}}\)
MathBot Answer:
\[\mathcal{L}_{s}^{-1}\left\{L - 1 s^{2} \cdot \frac{1}{\left(a + s\right)^{3}}\right\}(t) = \left(L e^{t \left(\operatorname{re}{\left(a\right)} + i \operatorname{im}{\left(a\right)}\right)} \delta\left(t\right) - \frac{a^{2} t^{2} \theta\left(t\right)}{2} + 2 a t \theta\left(t\right) - \theta\left(t\right)\right) e^{- t \left(\operatorname{re}{\left(a\right)} + i \operatorname{im}{\left(a\right)}\right)}\]
$e$ is Euler's number, a mathematical constant that is approximately $2.71828$.
$\theta$ is the Heaviside function.
$\delta$ is the Dirac delta function.