how do you i do the partial fraction for 1/(s^2)(s^2 +2s +3)
What you typed is really just
(s^2 +2s +3)/s^2
which is
= 1 + 2/s + 3/s^2
I don't know if this is what you needed.
If, however, you meant
1/((s^2)(s^2 +2s +3)) then that is
A/s + B/s^2 + (Cs+D)/(s^2+2s+3)
= A(s*(s^2+2s+3)) + B(s^2+2s+3) + (Cs+D)s^2
= As^3+2As^2+3As + Bs^2+2Bs+3B + Cs^3+Ds^2
= (A+C)s^3 + (2A+B+D)s^2 + (3A+2B)s + 3B
all over s^2(s^2+2s+3)
So, to make the two sides equal, we need
A+C = 0
2A+B+D = 0
3A+2B = 0
3B = 1
Solve that and you get
B = 1/3
A = -2/9
C = 2/9
D = 1/9
So, the partial fraction decomposition is
-2/9s + 1/3s^2 + (2s-1)/9(s^2+2s+3)
oops the first one should be s^2 +1
so, fix it and follow the steps.
If you enter your formula at wolframalpha.com, it will show the decomposition, so you can check your work. Watch to be sure you include necessary parentheses. wolframalpha will show how it interprets your input.
To perform partial fraction decomposition for the expression 1/(s^2)(s^2 + 2s + 3), follow these steps:
Step 1: Factorize the denominator
First, we factorize the quadratic expression s^2 + 2s + 3. Since the discriminant (2^2 - 4 * 1 * 3) is negative, the quadratic is irreducible. So, we cannot factorize it further.
Step 2: Express the rational expression in partial fractions
The expression can be written in the form A/s^2 + B/(s^2 + 2s + 3), where A and B are constants.
Step 3: Find the common denominator
To combine the fractions, we need to find the common denominator. In this case, the least common denominator (LCD) is (s^2)(s^2 + 2s + 3).
Step 4: Write the equations
Set up the equations based on the common denominator:
1 = A(s^2 + 2s + 3) + B(s^2)
Step 5: Solve for unknowns
Expand the equation and group like terms:
1 = (A + B) s^2 + (2A) s + (3A)
Since the terms on the left and right sides must be equal, the coefficients of like terms must be equal.
Matching the coefficients, we get the equations:
A + B = 0
2A = 0
3A = 1
From the second equation, we find A = 0. Since A = 0, the first equation becomes B = 0.
Step 6: Write the partial fraction expression
The partial fraction decomposition for 1/(s^2)(s^2 + 2s + 3) is:
1/(s^2)(s^2 + 2s + 3) = 0/s^2 + 0/(s^2 + 2s + 3)
Simplifying, we have:
1/(s^2)(s^2 + 2s + 3) = 0 + 0/(s^2 + 2s + 3)
Therefore, the partial fraction decomposition for the given expression is 0/s^2 + 0/(s^2 + 2s + 3).