General Contents

Detailed Contents

Programmed tutorial: Partial Fractions Example 6

If you find this page helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor

If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.

Find the partial fraction decomposition of

General Contents

Detailed Contents

Set up the partial fractions.

We get

Equation 1

How do we proceed?

We can determine A, B, C, and D by combining the terms on the right hand side.
What is the least common denominator for them?

s² * (s – 7) * (s – 1)

Combine these fractions so that we can equate the numerators.

Using the least common denominator, we get

Equation 2:

How can we solve for A, B, C, and D?

There are two methods:

a) We can take advantage of the fact that this equation must hold for all value of s. Accordingly, we can choose simplifying values of s, like s = 0, s = 1, and s = 7.

b) Alternatively, we can equate coefficients of like terms and solve four equations simultaneously.

We’ll do both as a check, and start with (a).
Substitute s = 0 in the numerator on both sides of equation 2.

Substitute s = 1 in the numerator on both sides of equation 2.

Substitute s = 7 in the numerator on both sides of equation 2.

Substitute s = 2 in the numerator on both sides of equation 2.

Substitute in this equation the values of B, C, and D just obtained.

Solve for A.

Now we can check these four values by using method (b).
First distribute the multiplication in the numerators of equation 2.

Collect like powers of s on the right-hand side.

Equate coefficients of like powers.

For

Equation 3
For

Equation 4
For

Equation 5
For

Equation 6

We have 4 equations in the 4 unknowns: A, B, and C.
How can we solve for them?

We can use the Substitution and Elimination Methods.

How should we proceed?

Solve for B in equation 6, then for A in equation 5.
After that, use those values to simplify equation 3 and 4.

Find B.

From equation 6, B = 1

Find A.

From equation 5 with B = 1 we get

Use these results to simplify equations 3 and 4.

We get

Add these resulting two equations to eliminate C.

We get

Find C.

Using one of the equations above, we get

In summary, we have agreement between our two methods.

Use these results to rewrite equation 1 above to express Y(s).

$partial fraction expansion of a linear over the product of 2 linears and the square of the independent variable$

The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor