How to Find the Laplace Transform of ( sqrt{t} e^{3t} )

Calculating the Laplace transform of f(t) sqrt{t} e^{3t} is an important skill in advanced calculus and differential equations. This guide will walk you through the process step by step, utilizing the definition of the Laplace transform and properties of the Gamma function.

1. Definition of the Laplace Transform

The Laplace transform of a function f(t) is given by:

[mathcal{L}{f(t)} int_{0}^{infty} e^{-st} f(t) , dt]

Substituting f(t) sqrt{t} e^{3t} into the definition, we have:

[mathcal{L}{sqrt{t} e^{3t}} int_{0}^{infty} e^{-st} sqrt{t} e^{3t} , dt int_{0}^{infty} sqrt{t} e^{3t - st} , dt]

Combining the exponent terms, we get:

[mathcal{L}{sqrt{t} e^{3t}} int_{0}^{infty} sqrt{t} e^{-st - 3t} , dt]

2. Recognizing the Gamma Function

The integral can be recognized as a form of the Gamma function. Let's rewrite the integral:

[int_{0}^{infty} t^{1/2} e^{-s - 3t} , dt]

Using the property of the Gamma function, we know:

[mathcal{L}{t^n e^{-beta t}} frac{n!}{(s beta)^{n 1}}]

Here, n frac{1}{2} and beta 3 . Thus:

[mathcal{L}{t^{1/2} e^{-3t}} Gammaleft(frac{3}{2}right) cdot frac{1}{(s 3)^{3/2}}]

3. Using the Gamma Function

The value of Gammaleft(frac{3}{2}right) is known to be:

[Gammaleft(frac{3}{2}right) frac{sqrt{pi}}{2}]

Substituting this value, we get:

[mathcal{L}{sqrt{t} e^{3t}} frac{frac{sqrt{pi}}{2}}{(s - 3)^{3/2}} frac{sqrt{pi}}{2(s - 3)^{3/2}}]

Therefore, the Laplace transform of sqrt{t} e^{3t} is:

[mathcal{L}{sqrt{t} e^{3t}} frac{sqrt{pi}}{2(s - 3)^{3/2}} quad text{for} ; s > 3]

4. Arranging the Work

For the Laplace transform of t^{alpha} , we use the formula:

[mathcal{L}{t^{alpha}} frac{Gamma(alpha 1)}{s^{alpha 1}}]

By taking alpha frac{1}{2} , we get:

[mathcal{L}{sqrt{t}} frac{Gamma(frac{3}{2})}{s^{3/2}} frac{sqrt{pi}}{2s^{3/2}}]

Using the linearity property of the Laplace transform, if mathcal{L}{f(t)} F(s) , then:

[mathcal{L}{e^{kt} f(t)} F(s - k)]

With k 3 and f(t) sqrt{t} , we get:

[mathcal{L}{sqrt{t} e^{3t}} frac{sqrt{pi}}{2(s - 3)^{3/2}}]

Conclusion

The Laplace transform of sqrt{t} e^{3t} is given by: