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### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# In Problems 11–20, determine the partial fraction expansion for the given rational function.$\frac{\mathbf{-}\mathbf{8}{\mathbf{s}}^{\mathbf{2}}\mathbf{-}\mathbf{5}\mathbf{s}\mathbf{+}\mathbf{9}}{\mathbf{\left(}\mathbf{s}\mathbf{+}\mathbf{1}\mathbf{\right)}\mathbf{\left(}{\mathbf{s}}^{\mathbf{2}}\mathbf{-}\mathbf{3}\mathbf{s}\mathbf{+}\mathbf{2}\mathbf{\right)}}$

The partial fraction expansions for the given rational function $\frac{-8{\mathrm{s}}^{2}-5\mathrm{s}+9}{\left(\mathrm{s}+1\right)\left({\mathrm{s}}^{2}-3\mathrm{s}+2\right)}$ is $\frac{1}{\mathrm{s}+1}-\frac{11}{\mathrm{s}-2}+\frac{2}{\mathrm{s}-1}$.

See the step by step solution

## Step 1: Definition of partial fraction expansion

Any number which can be easily represented in the form of $\mathrm{p}/\mathrm{q}$, such that localid="1662726245663" $\mathrm{p}$ and localid="1662726249923" $\mathrm{q}$ are integers and localid="1662726253823" role="math" $\mathrm{q}\ne 0$ is known as a rational number.

Similarly, we can define a rational function as the ratio of two polynomial functions $\mathrm{P}\left(\mathrm{x}\right)$ and $\mathrm{Q}\left(\mathrm{x}\right)$ , where localid="1662726263552" $\mathrm{P}$ and localid="1662726260804" $\mathrm{Q}$ are polynomials in localid="1662726257626" $\mathrm{x}$ and localid="1662726267995" $\mathrm{Q}\left(\mathrm{x}\right)\ne 0$.

A rational function is known as proper if the degree of localid="1662726271465" $\mathrm{P}\left(\mathrm{x}\right)$ is less than the degree of $\mathrm{Q}\left(\mathrm{x}\right)$; otherwise, it is known as an improper rational function.

With the help of the long division process, we can reduce improper rational functions to proper rational functions. Therefore, if $\text{P}\left(\text{x}\right)/\text{Q}\left(\text{x}\right)$ is improper, then it can be expressed as:

$\frac{\mathrm{P}\left(\mathrm{x}\right)}{\mathrm{Q}\left(\mathrm{x}\right)}{=}{\mathrm{A}}{\left(}{\mathrm{x}}{\right)}{+}\frac{\mathrm{R}\left(\mathrm{x}\right)}{\mathrm{Q}\left(\mathrm{x}\right)}$

Here,localid="1662726284688" $\mathrm{A}\left(\mathrm{x}\right)$ is a polynomial in localid="1662726275840" $\mathrm{x}$ and localid="1662726279992" $\mathrm{R}\left(\mathrm{x}\right)/\mathrm{Q}\left(\mathrm{x}\right)$ is a proper rational function.

## Step 2: Determine the partial fraction expansion for the given rational function

The given rational function is $\frac{-8{\mathrm{s}}^{2}-5\mathrm{s}+9}{\left(\mathrm{s}+1\right)\left({\mathrm{s}}^{2}-3\mathrm{s}+2\right)}$

Rewrite $\frac{-8{\mathrm{s}}^{2}-5\mathrm{s}+9}{\left(\mathrm{s}+1\right)\left({\mathrm{s}}^{2}-3\mathrm{s}+2\right)}$ as a sum of partial fractions as:

$\frac{-8{\mathrm{s}}^{2}-5\mathrm{s}+9}{\left(\mathrm{s}+1\right)\left(\mathrm{s}-2\right)\left(\mathrm{s}-1\right)}=\frac{\mathrm{A}}{\mathrm{s}+1}+\frac{\mathrm{B}}{\mathrm{s}-2}+\frac{\mathrm{C}}{\mathrm{s}-1}$

Multiply both sides by the LCD $\left(\mathrm{s}+1\right)\left(\mathrm{s}-2\right)\left(\mathrm{s}-1\right)$ as follows:

$-8{\mathrm{s}}^{2}-5\mathrm{s}+9=\mathrm{A}\left(\mathrm{s}-2\right)\left(\mathrm{s}-1\right)+\mathrm{B}\left(\mathrm{s}+1\right)\left(\mathrm{s}-1\right)+\mathrm{C}\left(\mathrm{s}+1\right)\left(\mathrm{s}-2\right)$

Find the constants as:

For $\mathrm{s}=-1,-8{\left(-1\right)}^{2}-5\left(-1\right)+9=\mathrm{A}\left(-3\right)\left(-2\right)⇒\mathrm{A}=1$

For $\mathrm{s}=2:-8{\left(2\right)}^{2}-5\left(2\right)+9=\mathrm{B}\left(3\right)\left(1\right)⇒\mathrm{B}=-11$.

For $\mathrm{s}=1:-8{\left(1\right)}^{2}-5\left(1\right)+9=\mathrm{C}\left(2\right)\left(-1\right)⇒\mathrm{C}=2$ .

Substitute the value of constants into $\frac{-8{\mathrm{s}}^{2}-5\mathrm{s}+9}{\left(\mathrm{s}+1\right)\left(\mathrm{s}-2\right)\left(\mathrm{s}-1\right)}=\frac{\mathrm{A}}{\mathrm{s}+1}+\frac{\mathrm{B}}{\mathrm{s}-2}+\frac{\mathrm{C}}{\mathrm{s}-1}$ as follows:

$\begin{array}{c}\frac{-8{\mathrm{s}}^{2}-5\mathrm{s}+9}{\left(\mathrm{s}+1\right)\left(\mathrm{s}-2\right)\left(\mathrm{s}-1\right)}=\frac{1}{\mathrm{s}+1}+\frac{-11}{\mathrm{s}-2}+\frac{2}{\mathrm{s}-1}\\ =\frac{1}{\mathrm{s}+1}-\frac{11}{\mathrm{s}-2}+\frac{2}{\mathrm{s}-1}\end{array}$

Therefore, the partial fraction expansion for the given rational function is $\frac{1}{\mathrm{s}+1}-\frac{11}{\mathrm{s}-2}+\frac{2}{\mathrm{s}-1}$.