The Method of Partial Fractions

These equations can be solved (although not easily) by integration by the method of partial fractions, by matrices, or by Laplace transforms. For the case where [I]o = [P]0 = 0, the concentrations are... 

Now we use the method of partial fractions to write the left side as sum of two simple terms. Let... 

After integration by the method of partial fractions, the result is ... 

Using the method of partial fractions the right side of equation (5.12) can be written as... 

This integral may be evaluated either by splitting into partial fractions, or by graphical or numerical means. Using the method of partial fractions, we obtain after some fairly lengthy manipulation ... 

The Together statement collects all terms of an expression together over a common denominator, while the Apart statement breaks the expression apart into terms with simple denominators, as in the method of partial fractions. The theorem of partial fractions states that if Q(x) can be factored in the form... 

The fundamental formula of the method of partial fractions is a theorem of algebra that says that if Q(x) is given by Eq. (5.45) and P x) is of lower degree than Q x), then... 

Now integrate the first equation by the method of partial fractions. Evaluate the integration constant for x = 0 when t — 0 and show that... 

Case 2. y%i y. In this case, the two reactants are present in an arbitrary ratio, not the required stoichiometric ratio. Using the method of partial fractions, we can rewrite... 

The method of partial fractions is described in elementary calculus texts. 

Solving this equation requires using the technique known as the method of partial fractions. The fraction on the left hand side of Eq. (2.6) can be separated into two fractions by separating the denominator as follows. 

The ordinary power representation has been chosen because the expansion of the generating function works with the method of partial fractions. The coefficient at gathers all the probabilities for a total degree of polymerization of P. Here again the... 

We proceed by the method of partial fractions (which is employed in the general case too), and look for the values of a. / . and y such that... 

Integrating Equation 2.113 using the method of partial fraction, we get... 

The integral on the right is simply k,t. The integral on the left is evaluated by using the method of partial fractions (see Mathematical toolkit 6.2) ... 

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