How to Find the Inverse Laplace Transform of ( frac{1}{s^2 s 2} )

The Inverse Laplace Transform is a key operation in the analysis and control of linear time-invariant systems, often used in electrical and mechanical engineering, as well as in solving differential equations. Here, we will guide you through finding the inverse Laplace transform of the function ( frac{1}{s^2 s 2} ).

1. Completing the Square

To solve the inverse Laplace transform, we first need to complete the square for the denominator.

tWrite the denominator in the form ((s a)^2 b^2): t t ttStart by adding and subtracting a constant to complete the square. In this case, add and subtract (frac{1}{4}): tt tt[ s^2 s 2 s^2 s frac{1}{4} - frac{1}{4} 2 ] ttSimplify the expression: tt tt[ s^2 s 2 left(s frac{1}{2}right)^2 frac{7}{4} ] ttThus, the function can be written as: tt tt[ frac{1}{s^2 s 2} frac{1}{left(s frac{1}{2}right)^2 frac{7}{4}} ] t t

2. Using the Table of Laplace Transforms

Using the table of Laplace transforms, the inverse transform of a function in the form (frac{b}{(s a)^2 b^2}) is given by:

tThe general form of the inverse Laplace transform is: t t tt tt[ mathcal{L}^{-1}left{frac{b}{(s a)^2 b^2}right} e^{-at} sin(bt)u(t) ] ttWhere (u(t)) is the unit step function, which is 0 for (t t t tCompare this form with our function: t t ttHere, (a frac{1}{2}) and (b sqrt{frac{7}{4}} frac{sqrt{7}}{2}). ttThus, the inverse Laplace transform is: tt tt[ mathcal{L}^{-1}left{frac{1}{(s frac{1}{2})^2 left(frac{sqrt{7}}{2}right)^2}right} e^{-frac{t}{2}} sinleft(frac{sqrt{7}}{2} tright) u(t) ] t t

Therefore, the final result is:

[ mathcal{L}^{-1}left{frac{1}{s^2 s 2}right} frac{2}{sqrt{7}} e^{-frac{t}{2}} sinleft(frac{sqrt{7}}{2} tright) u(t) ]

3. Detailed Steps

Here's a more detailed breakdown of the steps:

tComplete the square: t t tt tt[ s^2 s 2 left(s frac{1}{2}right)^2 frac{7}{4} ] t t tUse the inverse Laplace transform formula: t t tt tt[ mathcal{L}^{-1}left{frac{1}{left(s frac{1}{2}right)^2 left(frac{sqrt{7}}{2}right)^2}right} e^{-frac{t}{2}} sinleft(frac{sqrt{7}}{2} tright) u(t) ] t t tMultiply by the constant factor: t t tt tt[ mathcal{L}^{-1}left{frac{1}{s^2 s 2}right} frac{2}{sqrt{7}} e^{-frac{t}{2}} sinleft(frac{sqrt{7}}{2} tright) u(t) ] t t

Conclusion

By following this method, you can easily find the inverse Laplace transform of the function (frac{1}{s^2 s 2}). This technique is widely applicable and can be used in various engineering and scientific contexts where the Laplace transform is used.

Keywords

Laplace Transform, Inverse Laplace Transform, Unit Step Function