Chapter 3, Problem 3.9E

Interpretation Introduction

(a)

Interpretation:

The functions of time that will appear in $y\left(t\right)$ for the given function $Y\left(s\right)=\frac{2}{s\left(s^2+4s\right)}$ is to be determined. Also, whether the function $y\left(t\right)$ is oscillatory or has constant value for large values of t is to be determined.

Concept introduction:

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

$f\left(t\right)$ is calculated by taking inverse Laplace transform of the function $F\left(s\right)$.

PFE is the partial fraction expansion is the method of expanding the denominator of a fraction into simpler terms.

Interpretation Introduction

(b)

Interpretation:

The functions of time that will appear in $y\left(t\right)$ for the given function $Y\left(s\right)=\frac{2}{s\left(s^2+4s+3\right)}$ is to be determined. Also, whether the function $y\left(t\right)$ is oscillatory or have constant value for large values of t is to be determined.

Concept introduction:

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

$f\left(t\right)$ is calculated by taking inverse Laplace transform of the function $F\left(s\right)$.

PFE is the partial fraction expansion is the method of expanding the denominator of a fraction into simpler terms.

Interpretation Introduction

(c)

Interpretation:

The functions of time that will appear in $y\left(t\right)$ for the given function $Y\left(s\right)=\frac{2}{s\left(s^2+4s+4\right)}$ is to be determined. Also, whether the function $y\left(t\right)$ is oscillatory or have constant value for large values of t is to be determined.

Concept introduction:

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

$f\left(t\right)$ is calculated by taking inverse Laplace transform of the function $F\left(s\right)$.

PFE is the partial fraction expansion is the method of expanding the denominator of a fraction into simpler terms.

Interpretation Introduction

(d)

Interpretation:

The functions of time that will appear in $y\left(t\right)$ for the given function $Y\left(s\right)=\frac{2}{s\left(s^2+4s+8\right)}$ is to be determined. Also, whether the function $y\left(t\right)$ is oscillatory or have constant value for large values of t is to be determined.

Concept introduction:

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

$f\left(t\right)$ is calculated by taking inverse Laplace transform of the function $F\left(s\right)$.

PFE is the partial fraction expansion is the method of expanding the denominator of a fraction into simpler terms.

Interpretation Introduction

(e)

Interpretation:

The functions of time that will appear in $y\left(t\right)$ for the given function $Y\left(s\right)=\frac{2\left(s+1\right)}{s\left(s^2+4\right)}$ is to be determined. Also, whether the function $y\left(t\right)$ is oscillatory or have constant value for large values of t is to be determined.

Concept introduction:

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

$f\left(t\right)$ is calculated by taking inverse Laplace transform of the function $F\left(s\right)$.

PFE is the partial fraction expansion is the method of expanding the denominator of a fraction into simpler terms.