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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

01

Given information

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

02

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)

03

Properties used for given function

Formula used for calculations is :

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

04

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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Most popular questions from this chapter

Solve $(12.3)$ if $G = 0$ and $d G / d t = 0$ at $t = 0$ to obtain (12.5). Hint: Use L28 and L3 to find the inverse transform.

Using $(3.9)$ , find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after $(3.9)$ , and Example 1.

$y' c o s x + y = c o s^{2} x$

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

$y " + 16 y = 8 c o s 4, y_{0} = y_{0}^{'} = 0$

Find the inverse Laplace transform of $e^{- 2 P} l P^{2}$ in the following ways:

(a) Using L5 and L27 and the convolution integral of Section 10;

(b) Using L28.

$(1 + y^{2}) d x + x y d y = 0, y = 0$ when $x = 5$ .

See all solutions

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