I'm studying Laplace transformations for my differential equations class and typically there's a partial fraction decomposition involved, which can be very long and demanding for calculations by hand, if done the standard way.
I am aware of some of the tricks used to speed-up this procedure (like the usage of limits at infinity, or multiplying with denominators and taking particular values of $s$), however I am not able to apply them at this example:
$$\frac {s} {(s^2+2s+5)(s^2+4)} = \frac {\alpha s + \beta} {s^2+2s+5} + \frac {\gamma s + \delta} {s^2+4}$$
For example, if I attempt to extract a relation for $\gamma$ and $\delta$, by multiplying with $s^2+4$ and taking $s=2i$, I get and equation that involves complex numbers; that makes me feel I have not gained much in terms of number of operations.
Are there any better tricks for this example?
Best Answer
Let's apply the Heaviside cover-up method, nonlinear version, as described by Bill Dubuque.
$$\rm \frac{x}{(x^2\!+2x+5)(x^2\!+4)} \ =\ \frac{ax+b}{x^2\!+2x+5}\, +\ \frac{cx+d}{x^2\!+4}$$
Clearing denominators yields
$$\rm x\, =\, (x^2\!+4)(ax+b)\, +\, (x^2\!+2x+5)(cx+d) $$
Evaluating this mod $\rm\ x^2\! +\! 4\,\: $ i.e.$\:$ iteratively applying the rewrite rule $\rm\ x^2 \to -4\,\ $ yields
$$\rm x\, =\, (c\!+\!2d)\ x\,+\, d\!-\!8c\ \ \Rightarrow\ \ c = 1/17,\,\ d= 8/17$$
Evaluating it mod $\rm\ x^2\! +\! 2x\!+\!5\:,\: $ i.e.$\:$ iteratively applying rewrite rule $\rm\ x^2 \to -2x\!-\!5\: $ yields
$$\rm x\, =\, (3a\!-\!2b)\ x \,+\, 10a\!-\!b\ \ \Rightarrow\ \ a = -1/17,\,\ b = -10/17$$
That seems easy enough to me.