Question

# use properties of the Laplace transform and the table of Laplace transforms to determine L[f] f(t)=frac{e^{-5t}}{sqrt{t}}

Laplace transform
use properties of the Laplace transform and the table of Laplace transforms to determine L[f]
$$f(t)=\frac{e^{-5t}}{\sqrt{t}}$$

2020-12-31
$$\text{step 1}$$
$$\text{Given that: } f(t)=\frac{e^{-5t}}{\sqrt{t}}$$
$$\text{To solve the given, } f(t)=e^{-5t} \cdot t^{-\frac{1}{2}}$$
$$\text{To simplify the Laplace } t^{-\frac{1}{2}}$$
$$L\left[t^{-\frac{1}{2}}\right]=\bigg(\frac{\pi}{s}\bigg)^{\frac{1}{2}}$$
$$L\left[e^{at}\right]=\frac{1}{s-a}$$
$$\text{So , }$$
$$L\left[e^{-5t}\right]=\frac{1}{s+5}$$
$$\text{Step 2}$$
$$\text{Apply the first shiffting theorem with }a=-5$$
$$\text{By this theorem, }$$
$$L\left[e^{at} \cdot f(t)\right]=F(s-a)$$
$$\text{Then, }$$
$$L\left[e^{-5t} \cdot \bigg(t^{-\frac{1}{2}}\bigg)\right]=\bigg(\frac{\pi}{s+5}\bigg)^{\frac{1}{2}}$$
$$=\sqrt{\frac{\pi}{s+5}}$$
$$\text{the transform of the given function is } \sqrt{\frac{\pi}{s+5}}$$