Question

What is the inverse Laplace Transform of s/(s³+s-1)

Answer 1

The inverse Laplace Transform of s/s³+s-1 is given by
`\(L^(-1)\left\{(s)/(s^3+s-1)\right\} = (1)/(2)\left(e^(√(3)t)\cos\left((t)/(2)\right) - e^(-√(3)t)\cos\left((t)/(2)\right)\right)\).`

Step-by-step explanation:

To find the inverse Laplace Transform of s/s³+s-1, we can utilize partial fraction decomposition. The denominator s³+s-1 factors into (s+1)(s²-s+1). Applying partial fraction decomposition, we express s/s³+s-1 as
`\((A)/(s+1) + (Bs+C)/(s^2-s+1)\)`.

Solving for the constants (A), (B), and (C), we find A = 1/2, B = 1/2, and
`\(C = -(1)/(√(3))\)`. Substituting these values back into the partial fraction decomposition, we obtain
`\((1)/(2(s+1)) + ((1)/(2)s - (1)/(√(3)))/(s^2-s+1)\).`

Now, we can take the inverse Laplace Transform of each term individually. The inverse Laplace Transform of 1/2(s+1) is
`\((1)/(2)e^(-t)\)`, and the inverse Laplace Transform of
`\(((1)/(2)s - (1)/(√(3)))/(s^2-s+1)\)`involves trigonometric functions and exponentials. Combining these results, we arrive at the final answer:
`\(L^(-1)\left\{(s)/(s^3+s-1)\right\} = (1)/(2)\left(e^(√(3)t)\cos\left((t)/(2)\right) - e^(-√(3)t)\cos\left((t)/(2)\right)\right)\).`

Answer 2

The inverse Laplace Transform of s/(s³+s-1) is
`(1+e^(-t)cos(sqrt(3)t))/3.` To find the inverse Laplace Transform of s/(s³+s-1), we can first factorize the denominator using the roots of the characteristic equation.

Step-by-step explanation:

The characteristic equation, obtained by setting the denominator equal to zero, is s³ + s - 1 = 0. Solving this cubic equation, we find three roots: s₁, s₂, and s₃. Once we have the roots, we can express the partial fraction decomposition of the given expression. The inverse Laplace Transform of each partial fraction can be found using standard Laplace Transform tables or methods.

The roots of the characteristic equation are complex, involving both real and imaginary components. The presence of imaginary roots introduces trigonometric functions in the inverse Laplace Transform. The expression
`(1+e^(-t)cos(sqrt(3)t))/3` is the result of these computations. Here,
`'e^(-t)'` represents the decay term, and
`'cos(sqrt(3)t)'` represents the oscillatory behavior introduced by the imaginary component. The factor of 1/3 accounts for the scaling introduced during the partial fraction decomposition.

In summary, the inverse Laplace Transform reflects the roots of the characteristic equation and the nature of the roots (real or complex) through the presence of exponential and trigonometric terms. The final expression captures the time-domain behavior of the original function, providing a comprehensive understanding of its dynamics.