To find the partial fraction decomposition of:

$\frac{5s^2 + 22s + 14}{(s + 1)^2(s + 2)}$

Step-by-Step Breakdown to Find the Partial Fraction Decomposition

1. Set Up the Partial Fractions: Since the denominator is $(s + 1)^2(s + 2)$, the decomposition is:

   $\frac{5s^2 + 22s + 14}{(s + 1)^2(s + 2)} = \frac{A}{s + 1} + \frac{B}{(s + 1)^2} + \frac{C}{s + 2}$

2. Clear the Denominator: Multiply both sides by $(s + 1)^2(s + 2)$ to remove the fractions:

   $5s^2 + 22s + 14 = A(s + 1)(s + 2) + B(s + 2) + C(s + 1)^2$

3. Expand Each Term: Expand the right-hand side:

   • Expand $A(s + 1)(s + 2)$: $A(s^2 + 3s + 2)$

   • Expand $B(s + 2)$: $Bs + 2B$

   • Expand $C(s + 1)^2$: $C(s^2 + 2s + 1)$

   Combine these expanded terms:

   $5s^2 + 22s + 14 = A(s^2 + 3s + 2) + (Bs + 2B) + C(s^2 + 2s + 1)$

4. Combine Like Terms: Collect like terms on the right-hand side:

   $5s^2 + 22s + 14 = (A + C)s^2 + (3A + B + 2C)s + (2A + 2B + C)$

5. Set Up the System of Equations: Match the coefficients on both sides to create a system of equations:

   • Coefficient of $s^2$: $A + C = 5$
   • Coefficient of $s$: $3A + B + 2C = 22$
   • Constant term: $2A + 2B + C = 14$

6. Solve for $A$, $B$, and $C$:

   • From the first equation: $C = 5 - A$.

   • Substitute $C = 5 - A$ into the second and third equations:

     Equation for $s$ coefficient: $3A + B + 2(5 - A) = 22$ Simplify: $3A + B + 10 - 2A = 22$ $A + B = 12$

     Equation for constant term: $2A + 2B + (5 - A) = 14$ Simplify: $A + 2B + 5 = 14$ $A + 2B = 9$

   • Solve the system:

     • From $A + B = 12$ and $A + 2B = 9$:

       Subtract $A + B = 12$ from $A + 2B = 9$: $(A + 2B) - (A + B) = 9 - 12$ $B = -3$

     • Substitute $B = -3$ into $A + B = 12$: $A - 3 = 12$ $A = 15$

     • Find $C$ using $C = 5 - A$: $C = 5 - 15 = -10$

Final Answer for the Partial Fractions

$\frac{5s^2 + 22s + 14}{(s + 1)^2(s + 2)} = \frac{15}{s + 1} - \frac{3}{(s + 1)^2} - \frac{10}{s + 2}$

Explanation of How This Was Found

The answer was found by:

• Setting up the decomposition based on the factors of the denominator.
• Equating coefficients to form a system of linear equations.
• Solving the system to find the constants $A$, $B$, and $C$.

Would you like further clarification on any part of this process, or help with the next part of the problem?

Follow-up Questions

1. What are the implications if one of the terms in the denominator is repeated more than twice?
2. How do you check the correctness of your partial fraction decomposition?
3. What if the degree of the numerator was higher than the degree of the denominator?
4. How can you handle complex roots in partial fraction decomposition?
5. What role do initial conditions play in inverse Laplace transformations?

Tip

When solving for coefficients in partial fraction decomposition, verifying by substituting back into the original expression helps confirm the accuracy of your solution.