Partical Fractions, Integration and Equating coefficients

In summary, the conversation is about splitting a function into partial fractions. The person starts by factoring the denominator and rewriting the function in a different form. They then use substitution to solve for the coefficients and end up with a final answer of (-w^2 -w - 1) / (w^3) + 1/(w-1). However, they have doubts about their solution and ask for help.
  • #1
zodiacbrave
11
0

Homework Statement



Split the function into partial fractions. 1/(w^4-w^3)

Homework Equations



1/(w^4-w^3)

The Attempt at a Solution



I started by factoring the denominator to w^3(w-1) and re-writing the original function as

(Aw^2+Bw+C)/w^3 + D/(w-1) and set it = 1/(w^3(w-1))

I end up with 1 = (Aw^2+Bw+C)(w-1)+Dw^3

if I set w = 0 then,

-1 = C

if i set w = 1 then,
1 = D

then I start organizing everything and I end up with,

1 = [A + D]w^3 + [B-A]w^2 + [C-B]w - C

so,

0 = A + D
0 = B - A
0 = C - Bsince I know what D and C are,

A = -1

B = -1

so my final answer is (-w^2 -w - 1) / (w^3) + 1/(w-1)The book gives me a different answer.. I am pretty sure I messed up, probably at the start with factoring, can someone please help?

The book gives the answer to be 1/(w-1) -1/w - 1/w^2 - 1/w^3

Thank you
 
Last edited:
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  • #2
zodiacbrave said:
The book gives me a different answer..
What was the book's answer? Are you sure it's actually different?

I am pretty sure I messed up, probably at the start with factoring, can someone please help?
It's easy enough to check -- simply plug in a few values of w and see if your solution has the same value as the original fraction. Of course, this isn't a proof*, but if you made a mistake, it's extremely likely you'll catch it this way...


*: Actually, it is a proof if you do it right and know the relevant theorems. But I digress...
 

Related to Partical Fractions, Integration and Equating coefficients

1. What are particle fractions?

Particle fractions refer to the individual components that make up a mixture or substance. These particles can be solids, liquids, or gases and can be separated using various techniques such as filtration or distillation.

2. How is integration used in science?

Integration is used in science to calculate the area under a curve, which can represent various physical quantities such as displacement, velocity, or concentration. It is also used to determine the total amount of a substance present in a given sample.

3. What is the significance of equating coefficients?

Equating coefficients is important in chemistry and physics when balancing chemical equations or solving equations with multiple variables. It involves setting two expressions equal to each other and equating the coefficients of similar terms on both sides of the equation.

4. Can particle fractions be converted into each other?

Yes, particle fractions can be converted into each other through physical or chemical processes. For example, a solid can be melted into a liquid, and a liquid can be evaporated into a gas. Similarly, a chemical reaction can produce different particle fractions from the initial substances.

5. How is integration used in particle size analysis?

Integration is used in particle size analysis to determine the size distribution of particles in a sample. By measuring the intensity of scattered light at different angles, the area under the curve can be calculated using integration, providing information about the size and concentration of particles in the sample.

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