# Partial Fraction Expansion - Repeated Roots Case

## Homework Statement

Find Partial Fraction Expansion

10/[s (s+2)(s+3)^2]

## The Attempt at a Solution

10/[s (s+2)(s+3)^2] = A/s + B/(s+2) + C/(s+3)^2 + D/(s+3)

A = 10/[(s+2)(s+3)^2], s approaches 0 = 10/(2*3^2) = 5/9

B = 10/[s (s+3)^2], s approaches -2 = 10/(-2) = -5

C = 10/[s (s+2)], s approaches -3 = 10/[(-3)(-3+2)] = 10/[(-3)(-1)] = 10/3

For D,
First I find the equation that isolated C by multiply both sides by (s+3)^2

10/[s (s+2)] = [A(s+3)^2]/s + [B(s+3)^2]/(s+2) + C + D(s+3)

I then differentiate both sides with respect to s to find D? I have solved similar problems before with three terms, one repeated root, and to find the last constant I had to something similar to above and then differentiate both sides, but that doesn't seem to work in this case with four terms, one repeated root.

Any help would be appreciated thanks.

SammyS
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## Homework Statement

Find Partial Fraction Expansion
10/[s (s+2)(s+3)^2]

## Homework Equations

3. The Attempt at a Solution [/B]
10/[s (s+2)(s+3)^2] = A/s + B/(s+2) + C/(s+3)^2 + D/(s+3)

A = 10/[(s+2)(s+3)^2], s approaches 0 = 10/(2*3^2) = 5/9

B = 10/[s (s+3)^2], s approaches -2 = 10/(-2) = -5

C = 10/[s (s+2)], s approaches -3 = 10/[(-3)(-3+2)] = 10/[(-3)(-1)] = 10/3
That all looks fine.
For D,
First I find the equation that isolated C by multiply both sides by (s+3)^2

10/[s (s+2)] = [A(s+3)^2]/s + [B(s+3)^2]/(s+2) + C + D(s+3)

I then differentiate both sides with respect to s to find D? I have solved similar problems before with three terms, one repeated root, and to find the last constant I had to something similar to above and then differentiate both sides, but that doesn't seem to work in this case with four terms, one repeated root.

Any help would be appreciated thanks.
Now to find a value for D: Substitute some number for s, ##\ \text s = -1\ ## works well, then using your values for A, B, and C, solve for D.