Laplace Transforms

Question : Use initial and final value theorems to find the values of f(0) and f(∞) for F\left(s\right)=\frac{3+4s^2+5s}{s\left(s^2+4s+2\right)} Solution : We shall Use initial value theorem to find f(0) for…

Question : Use final value theorem to find f(∞) for \[F\left(s\right)=\frac{3+4s^2+5s}{s\left(s^2+4s+2\right)}\] Solution : Use final value theorem to find f(∞) for \[F\left(s\right)=\frac{3+4s^2+5s}{s\left(s^2+4s+2\right)}\] We need to write the function sF(s) as…

Question : Use initial value theorem to find f(0) for F(s)=(3+4s^2+5s)(s(s^2+4s+2)) Solution : We shall Use initial value theorem to find f(0) for F(s) = F(s)=(3+4s^2+5s)(s(s^2+4s+2)) Given function \[F\left(s\right)=\frac{3+4s^2+5s}{s\left(s^2+4s+2\right)}\] We…

Question : Find Inverse Laplace transform of Y(s)=1/((s^2+7s+9)) Solution : To Find Inverse Laplace transform of Y(s)=1/((s^2+7s+9)) Given function \[Y\left(s\right)=\frac{1}{(s^2+7s+9)}\] It can be written as \[Y\left(s\right)=\frac{1}{\left(s+\frac{7}{2}\right)^2-\frac{13}{4}}\] \[Y\left(s\right)=\frac{1}{\left(s+\frac{7}{2}\right)^2-\left(\sqrt{\frac{13}{2}}\right)^2}\] \[Y\left(s\right)=\frac{1}{\left(s+\frac{7}{2}-\sqrt{\frac{13}{2}}\right)\left(s+\frac{7}{2}+\sqrt{\frac{13}{2}}\right)}\] So the…

Question : Find Inverse Laplace transform of X(s)=(s-3+e^(-2s))/(s^2+4) Solution : To Find Inverse Laplace transform of X(s)=(s-3+e^(-2s))/(s^2+4) \[X\left(s\right)=\frac{s-3+e^{-2s}}{s^2+4}\] We can write function X(s) as \[X\left(s\right)=\frac{s}{s^2+4\ }-\frac{3}{s^2+4\ }+\frac{e^{-2s}}{s^2+4}\] First we shall…

Question : Find Inverse Laplace transform of F(s)= s/((s+7)(s^2+4)) using convolution theorem Solution : To Find Inverse Laplace transform of F(s)= s/((s+7)(s^2+4)) using convolution theorem Given \[\Rightarrow\ \ F\left(s\right)=\frac{1}{\left(s+7\right)}\frac{s}{(s^2+4)}\] Applying…

Question : Find Inverse Laplace transform of F(s)=(s-2)/(s^2+ω^2 ) Solution : To Find Inverse Laplace transform of F(s)=(s-2)/(s^2+ω^2 ) Partial fraction of F(s) is given as \[F\left(s\right)=\frac{s-2}{s^2+\omega^2}\] It can be…

Question : find inverse Laplace transform of 1/(s^2+2s-3) Solution: To find inverse Laplace transform of 1/(s^2+2s-3) Let it can be written as \[F\left(s\right)=\frac{1}{(s-1)(s+3)}\] and therefore \[F\left(s\right)=\left[\frac{1}{\left(s-1\right)}-\frac{1}{\left(s+3\right)}\right]\] Now apply Inverse Laplace…