Chapter 8: Q21P (page 439)

Use L29 and L11 to obtain $L (t e^{- a t} s i n b t)$ which is not in the table.

Short Answer

Expert verified

Answer

The solution is $L \{t e^{- a t} \sin b r\} = \frac{2 b (p + a)}{{({(w + a)}^{2} + b^{2})}^{2}}$

Step by step solution

Given information

A function as $f (t) = t e^{a t} \sin b t$

Definition of Laplace Transformation

A transformation of a function f(x) into the function g(t) that is useful especially in reducing the solution of an ordinary linear differential equation with constant coefficients to the solution of a polynomial equation.

The inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t)

Properties used for given function

Formula used for calculations is :

$L (29) : L \{e^{- B_{g} (t)}\} - G (p + a) L (11) : L \{t \sin a t\} = \frac{24 p}{{(p^{2} + ω^{2})}^{2}}$

Proof for given function

From (L29) $L \{e^{- d t} g (t)\} = G (p + a)$

Since from (L11) $L \{t \sin b t\} = \frac{2 h}{{(p^{2} + b^{2})}^{2}}$

Consider $g (t) = t \sin b t$ and by use of (L29),

$L \{e^{- a t_{g} (t)}\} = G (p + a)$ ;

$L \{t e^{a t} \sin b t\}$ Can be calculated by replace p by $(p + a)$ in equation(1) as,

$L \{t \sin b t\} = \frac{2 k p}{{(p^{2} + b^{2})}^{2}} L \{L_{e - a t \sin b t}\} = \frac{2 k (p + a)}{{({(p + a)}^{2} + b^{2})}^{2}}$

Hence $L \{t e^{- a t} \sin b t\} = \frac{2 b (p + a)}{{({(p + a)}^{2} + b^{2})}^{2}}$

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Use L29 and L11 to obtain $L (t e^{- a t} s i n b t)$ which is not in the table.

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Properties used for given function

Proof for given function

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Use L29 and L11 to obtain L(te-atsinbt)which is not in the table.

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Properties used for given function

Proof for given function

One App. One Place for Learning.

Most popular questions from this chapter

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Use L29 and L11 to obtain $L (t e^{- a t} s i n b t)$ which is not in the table.