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
ANSWERED
asked 2020-12-30
use properties of the Laplace transform and the table of Laplace transforms to determine L[f]
\(f(t)=\frac{e^{-5t}}{\sqrt{t}}\)

Answers (1)

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