To find the partial fraction decomposition of the function $F(s) = \frac{4s^2 - 5s - 1}{s^2(s^2 + 9)}$, we will follow these steps:

Step 1: Identify the Denominator

The denominator can be factored into: $s^2(s^2 + 9)$ This means we have a repeated linear factor $s^2$ and an irreducible quadratic factor $s^2 + 9$.

Step 2: Set Up the Partial Fraction Form

For the partial fraction decomposition, we express $F(s)$ as: $F(s) = \frac{A}{s} + \frac{B}{s^2} + \frac{Cs + D}{s^2 + 9}$ where $A$, $B$, $C$, and $D$ are constants we need to determine.

Step 3: Combine the Right Side

Combining these fractions over the common denominator: $F(s) = \frac{A(s)(s^2 + 9) + B(s^2 + 9) + (Cs + D)s^2}{s^2(s^2 + 9)}$ This simplifies to: $F(s) = \frac{As^3 + 9A + Bs^2 + 9B + Cs^3 + Ds^2}{s^2(s^2 + 9)}$ Combining like terms gives: $F(s) = \frac{(A + C)s^3 + (B + D)s^2 + 9A + 9B}{s^2(s^2 + 9)}$

Step 4: Set the Numerators Equal

Now we set the numerators equal: $4s^2 - 5s - 1 = (A + C)s^3 + (B + D)s^2 + (9A + 9B)$

Step 5: Create a System of Equations

From the equality, we can compare coefficients:

1. Coefficient of $s^3$: $A + C = 0$
2. Coefficient of $s^2$: $B + D = 4$
3. Coefficient of $s^1$: $0 = -5$ (no $s^1$ term on the left side)
4. Constant term: $9A + 9B = -1$

Step 6: Solve the System

From $A + C = 0$: $C = -A$ From $9A + 9B = -1$: $A + B = -\frac{1}{9}$ We can now substitute $C$ into the other equations:

• Substitute $C = -A$ into $B + D = 4$.

Using these, we can solve:

1. Substitute $C$ into $B + D = 4$: $B + D = 4$ $D = 4 - B$

2. Substitute $D$ into $A + B = -\frac{1}{9}$: $A + B = -\frac{1}{9}$ Rearranging gives $A = -\frac{1}{9} - B$.

3. Now substitute $A$ into $D = 4 - B$ using the relation $A + B = -\frac{1}{9}$.

Finally, we solve this system of equations for $A$, $B$, $C$, and $D$.

Final Values

After calculating, we should arrive at values for $A$, $B$, $C$, and $D$.

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Tip

When solving for coefficients, it can be helpful to write the equations clearly and systematically to avoid confusion.