Why does root multiplicity change the factors in partial fractions and homogenous differential equations? Why does a root with multiplicity larger than one generate different terms in partial fraction expansions and in finding the solution of homogeneous linear differential equations?
For example, consider the following expansion:
$$\frac{1}{s^2 - 4s + 4} = \frac{A}{s - 2} + \frac{B}{(s - 2)^2}$$
I can see why the denominators in the terms must be different, but not where the exponent comes from. I was taught this as "truth", but I don't know how to get there myself, or how to get a general formula.
The same is true for:
$$y'' - 4y' + 4y = 0$$
Leading to a general solution:
$$c_1 e^{2t} + c_2 te^{2t}$$
I don't know how the t appears and how what to do if the root's multiplicity was 3 (for example).
 A: As the nice comments eluded to, partial fractions are wrapped up in the Residue Theorem and Laurent Series.
Every function of the form $\displaystyle \frac{f(s)}{g(s)}$, where $f(s)$ and $g(s)$ are polynomials in $s$, can be reduced into the sum of fractions such that the denominator of each new fraction is either first degree or a quadratic polynomial raised to some power.
To each factor $g(s)$ of the form $(s-a)^m~$, assign a sum of $m$ fractions, of the form:
$$\displaystyle \frac{A_1}{s-a} + \frac{A_2}{(s-a)^2}+\cdots + \frac{A_m}{(s-a)^m}$$
To each factor, $g(s)$ of the form $(s^2+bs+c)^p$, assign a sum of partial fractions, of the form:
$$\displaystyle \frac{B_1s+C_1}{(s^2+bs+c)^1} + \frac{B_2s+C_2}{(s^2+bs+c)^2} + \cdots + \frac{B_ps + C_p}{(s^2+bs+c)^p}$$
where $A_i, B_j$ and $C_k ~(i=1, 2, \ldots, m;~j, k= 1, 2, \ldots , p)$ are constants which must be determined.
Set the original fraction, $\displaystyle \frac{f(s)}{g(s)}$, equal to the sum of the new fractions from above. Clear the resulting equation of fractions, and then equate coefficients of like powers of $s$, thereby resulting in a set of simultaneous linear equations in the unknown constants $A_i, B_j$ and $C_k$. Finally, solve for these $A_i, B_j$ and $C_k$.
Theory


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*Residue Theorem - Residue Calculus

*Residue and Laurent

*Residue Method for Partial Fractions Decomposition
Useful Resources


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*Nice little step-by-step JavaScript

*Wiki Partial Fractions

*Partial fractions decomposition - Methods Survey
