Partial Fractions

I am a great fan of yours and your work is excellent.

I was watching your video on Laplace Transform. I noted you took the STANDARD method for partial fractions which works in all cases. However, the partial fractions can be solved fairly quickly in certain cases.

This refers to 1/(s²+1)(s²+4).

1. Make sure the coefficients of s² are the same in both brackets to make sure they cancel out during subtraction.
2. Then find the difference between the constant terms. In this case, it is 4-1 = 3.
3. Bring that difference into the numerator

1/(s²+1)(s²+4) = 1/3 * 3/(s²+1)(s²+4)  = 1/3* (4-1) /(s²+1)(s²+4) = 1/3* (s²+4) - (s²+1) /( s²+1)(s²+4)

=1/3* (1/(s²+1)  - 1/(s²+4))

This is much faster and less prone to errors.

If the coefficients of s² are not equal, they can be equalized by taking a constant out of the bracket.

Similarly for 1/(s+1)(s+4)

If there is an s in the numerator, make the constant terms equal and get the difference between the ‘s’ terms in the numerator.

s/(s+1)(s+4) =4* s/(4s+4)(s+4) =4/3 * 3s/(4s+4)(s+4) =  4/3 * (4s+4) – (s+4) / (4s+4)(s+4) = 4/3 * (1/(s+4) – 1/4(s+1)

Hope you can share this with your viewers.