Chapter 8: Q2P (page 464)

Problems 2 and 3, use (12.6) to solve (12.1) when $f (t)$ is as given. $f (t) = s i n ω t$

Short Answer

Expert verified

Answer

The value of function $y (t) - \int_{0}^{t} \sin ω (t - t') f (t') d l$ is equal to $y (t) = \frac{1}{2 ω^{2}} [\sin (ω t) - ω t \cos ω t]$

Step by step solution

Given information

The given expressions are $f (t) = \sin ω t$ .

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)

Solve the given function

Use the function.

$f (t) = \sin ω t$

And then

$y (t) = \int_{0}^{l} \sin ω (t - t') f (t') d t y (t) = \int_{0}^{l} \sin ω (t - t') (\sin ω t) d t y (t) = \frac{1}{2 ω} \int_{0}^{l} 2 \sin (ω t - ω t) \cdot \sin ω t d t y (t) = \frac{1}{2 ω} [\int_{0}^{l} 2 \sin (ω t - ω t' - ω t') - \cos (ω t - ω t - ω t)] d t'$

Solve further

$y (t) = \frac{1}{2 ω} [\int_{0}^{t} {\{\frac{1}{- 2 ω} \sin (ω t - 2 ω t') d t\}}_{t - 0}^{t} - \cos' \int_{0}^{t} d t'] y (t) = \frac{1}{2 ω} [\int_{0}^{t} \frac{1}{- 2 ω} (\sin (ω t - 2 ω t') - \sin ω t - 0) - \cos {\{t\}}_{t - 0}^{t}] y (t) = \frac{1}{2 ω t^{2}} [\sin (ω t) - ω t \cos ω t]$

Thus, the value of function $y (t) = \int_{0}^{t} \sin (t - t') f (t') d t$ is equal to $y (t) = \frac{1}{2 ω^{2}} [\sin (ω t) - ω t \cos ω t]$ .

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Problems 2 and 3, use (12.6) to solve (12.1) when $f (t)$ is as given. $f (t) = s i n ω t$

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Solve the given function

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Problems 2 and 3, use (12.6) to solve (12.1) when f(t)is as given.f(t)=sinωt

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Solve the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Space Physics

Astrophysics

Further Mechanics and Thermal Physics

Kinematics Physics

Linear Momentum

Study anywhere. Anytime. Across all devices.

Problems 2 and 3, use (12.6) to solve (12.1) when $f (t)$ is as given. $f (t) = s i n ω t$