Hello, I am in a differential equations class need to find the inverse Laplace of
, but have forgotten how to do partial fractions with quadratic factors. What I got so far:
Letting s=0 I get , but where do I go from there?
Equate coefficients of the various powers of across the identity.
The number of on the LHS have to equal the number of on the RHS.
That gives you
Then equate coefficients of to get the value of
Alternatively you could substitute two other values for and solve the resulting simultaneous equations.
You have which is to be true for all s. So take s to be some simple numbers.
If s= 0, then . Putting that value into the equation, or .
Taking s= 1, that becomes 1/5+ B+ C= 0 or B+ C= -1/5. Taking s= -1, it is 1/5+ B- C= 0 or B- C= 1/5. Adding those, 2B= 0 so B= 0. Subtracting, 2C= -2/5 so C= -1/5.
Another way to get that is to multiply out . Now, because that is true for all s, we must have "corresponding coefficients" equal: A+ B= 0, C= 0, 5A= 1. That gives A= 1/5 and then (1/5)+ B= 0 so B= -1/5.