Question

In Problems 43-46 use Problems 41 and 42 and the fact that $\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }$to find the Laplace transform of the given function.

45.$f\left(t\right)=t^{3/2}$

Short answer

$\mathcal{L}\left\{f\right(t\left)\right\}=\frac{3\sqrt{\pi }}{4s^{\frac{5}{2}}}$

Step-by-step solution

Definition of Laplace Transform

Let f be a function defined for $t⩾0$ . Then the integral

$\mathcal{L}\left\{f\right(t\left)\right\}={\int }_0^x{e}^{-xt}f\left(t\right)dt$

is said to be the Laplace transform of f, provided that the integral converges.

Use Laplace Transform

Take Laplace transform of both side.

$\mathcal{L}\left\{f\right(t\left)\right\}=\mathcal{L}\left\{t^{\frac{3}{2}}\right\}\left(\mathcal{L}\left\{t^{\alpha }\right\}=\frac{\Gamma (\alpha +1)}{s^{\alpha }1}\right)$

$=\frac{\Gamma \left(\frac{3}{2}+1\right)}{s^{\frac{3}{2}+1}}\left(\Gamma \right(\alpha +1)=\alpha \Gamma (\alpha \left)\right)$

$=\frac{\frac{3}{2}\Gamma \left(\frac{3}{2}\right)}{s^{\frac{5}{2}}}$

$=\frac{\frac{3}{2}\Gamma \left(\frac{1}{2}+1\right)}{s^{\frac{5}{2}}}\left(\Gamma \right(\alpha +1)=\alpha \Gamma (\alpha \left)\right)$

$$\begin{gathered} =\frac{\frac{3}{2}·\frac{1}{2}\Gamma \left(\frac{1}{2}\right)}{s^{\frac{5}{2}}}\left(\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }\right) \\ =\frac{3\sqrt{\pi }}{4s^{\frac{5}{2}}} \end{gathered}$$

Therefore,$\mathcal{L}\left\{f\right(t\left)\right\}=\frac{3\sqrt{\pi }}{4s^{\frac{5}{2}}}$