I made this question, just wanted to make sure it didn't drown under the tide...
I've set a problem up, something like this.
s^4: A+D=0
s^3: -2A+B-3D+E=0
s^2: 2A-B+C+3D-3E=0
s^1: -2A+B-D+3E=4
s^0: A+C-E=4
It's easier to substitute for s the values that make the denominators zero and then demand equalty. You then get simpler equations than comparing the coefficients of equal powers of s on both sides.
Would anybody mind showing it out? I tried what Count Iblis said, but it didn't really work out well...
The paragraph after the thing with s as a variable is copy and pasted from Count Iblis.
Hi David,
If I give you the answer (I just verified that it is correct):
(s+1)/[(s^2+1)(s-1)^3] =
1/(s-1)^3 - 1/2 1/(s-1)^2 +
1/2 1/(s^2+1)
Can you verify that adding up the partial fractions yields the original fraction?
Can you extract your A, B, C, etc. from the solution and verify that it satisfies your equations?
Now, you have to admit that just writing down the equations by equating equal powers of s and solving them is not an effient way to solve the problem. It is perhaps a good algebra exercise, but I can assure you that people like me who have to do this for their work usually do not use this awkward method. only in somple cases when you have just two terms is the method useful.
There is a slight shortcut when you solve partial fractions by introducing the unknown constants: You add up the partial fractions so that you have a common denominator and then equate the two numerators as you've done.
You can insert s = 1, s = i and s = -i and demand that the two numerators are the same for these three cases. If you do that then you lose some terms so the equations becomes simpler Also, you can demand that the first and second derivatives of both sides are the same at s = 1.
I'm confused.
As soon as I get the so called 'Basic Equation,' I just don't know what to do after....
To solve this problem using substitution, we need to determine the values of s that make the denominators zero. Let's go step by step:
1. Set each denominator to zero and solve for s individually:
a) s^4: Since there is no denominator, we don't need to do anything.
b) s^3: Set the denominator (-2) equal to zero: -2 = 0.
Since this equation has no solution, there is no restriction on s.
c) s^2: Set the denominator (2) equal to zero: 2 = 0.
Again, this equation has no solution, so no restriction on s.
d) s^1: Set the denominator (-2) equal to zero: -2 = 0.
Similar to the previous cases, no restriction on s.
e) s^0: Set the denominator (1) equal to zero: 1 = 0.
Once again, no solution, no restriction on s.
In summary, there are no specific values for s that make the denominators zero, so we can continue without any substitution.
2. Now, we can equate the coefficients of equal powers of s on both sides:
a) s^4: A + D = 0
b) s^3: -2A + B - 3D + E = 0
c) s^2: 2A - B + C + 3D - 3E = 0
d) s^1: -2A + B - D + 3E = 4
e) s^0: A + C - E = 4
These equations represent the relationships between the coefficients A, B, C, D, and E.
3. Now, we can solve this system of equations using various methods such as substitution, elimination, or matrix methods. Let's use substitution for this example:
- From equation (a): A = -D
- Substitute A in equation (b): -2(-D) + B - 3D + E = 0
Simplify: 2D + B - 3D + E = 0
Rearrange: -D + B + E = 0 (equation f)
- Substitute A in equation (d): -2(-D) + B - D + 3E = 4
Simplify: 2D + B - D + 3E = 4
Rearrange: D + B + 3E = 4 (equation g)
- Substitute A in equation (e): (-D) + C - E = 4
Simplify: -D + C - E = 4
Rearrange: C - D - E = 4 (equation h)
Now, we have three equations (f), (g), and (h) that involve only D, B, C, and E.
4. Solve this new system of equations (f), (g), and (h) to find the values of D, B, C, and E. Once you have these values, substitute them back into the original equations (a) to (e) to obtain the corresponding values of A.
The solution to this system of equations will give you the values of A, B, C, D, and E that satisfy all the given equations.
I hope this explanation helps you understand how to approach the problem and solve it using substitution.