Question

Operate the Laplace Transform on the following text{(i) } f(t)=cos(3t) text{(ii) } f(t)=t^{frac{1}{2}}

Laplace transform
ANSWERED
asked 2021-01-28
Operate the Laplace Transform on the following
\(\text{(i) }\ f(t)=\cos(3t)\)
\(\text{(ii) }\ f(t)=t^{\frac{1}{2}}\)

Answers (1)

2021-01-29
\(\text{Step 1}\)
\(\text{to find the Laplace transform of }\ f(t)=\cos(3t) \ \text{proceed as follows}\\ L\left\{f(t)\right\}=L\left\{\cos(3t)\right\}\)
\(\text{using the formula }\ L\left\{\cos(at)\right\}=\frac{s^2}{s^2+a^2}\)
\(L\left\{\cos(3t)\right\}=\frac{s^2}{s^2+3^2}\)
\(=\frac{s^2}{s^2+9}\)
\(\text{therefore, }\ L\left\{\cos(3t)\right\}=\frac{s^2}{s^2+9}\)
\(\text{Step 2}\)
\(\text{to find the Laplace transform of }\ f(t)=t^{\frac{1}{2}} \ \text{proceed as follows}\)
\(L\left\{f(t)\right\}=L\left\{t^{\frac{1}{2}}\right\}\)
\(\text{using the formula} L\left\{t^{n-{\frac{1}{2}}}\right\}=\frac{(2n-1)!\sqrt{\pi}}{2^{n}s\frac{n+1}{2}}\)
\(\text{put } n=1\ \text{to get }\ L\left\{f(t)\right\}=L\left\{t^{\frac{1}{2}}\right\}\)
\(L\left\{t^{1-{\frac{1}{2}}}\right\}=\frac{(2(1)-1)!\sqrt{\pi}}{2^{1}s^{n+\frac{1}{2}}}\)
\(=\frac{(1)!\sqrt{\pi}}{2s^{1+\frac{1}{2}}}\)
\(=\frac{(1)!\sqrt{\pi}}{2s^{\frac{3}{2}}}\)
\(=\frac{\sqrt{\pi}}{2s^{\frac{3}{2}}}\)
\(\text{therefore, }\ L\left\{t^{\frac{1}{2}}\right\}=\frac{\sqrt{\pi}}{2s^{\frac{3}{2}}}\)
0
 
Best answer

expert advice

Have a similar question?
We can deal with it in 3 hours
...