How Do I Decompose This Fraction in My ODE?

In summary, the conversation is about solving an ODE involving substitution, specifically decomposing a fraction. The individual is struggling with breaking down the fraction into partial fractions and seeks advice. Another person suggests splitting the fraction into two parts and using substitution and arctan to solve it. The conversation concludes with the realization that the fraction may not even be able to be decomposed using partial fractions.
  • #1
smallzilla
2
0
Hello I am stuck on an ODE involving substitution. I have done the correct substitutions, but have become stuck on decomposing the fraction.
i have the following

∫(1/x)dx + ∫(u+1)/(u^2+1)du = 0

Im stuck on breaking the u down into a partial decomposition. Could anyone offer some advice on how to start decomposing this bad boy?

Thanks
 
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  • #2
You don't need to decompose that with partial fractions.
Split up the (u+1)/(u^2+1) as u/(u^2 + 1) + 1/(u^2+1).
Now you can do a substitution on the 1st term and the 2nd term is just arctan(u).
I'm not even sure that you could do a partial fraction decomposition on that because you can't factor the denominator.
 
  • #3
I see, thanks for pointing that out! I'm do for an algebra review it seems :)
 

FAQ: How Do I Decompose This Fraction in My ODE?

What is partial fraction decomposition?

Partial fraction decomposition is a method used to decompose a rational function into simpler fractions. This is helpful for solving integration problems or simplifying complex expressions.

When is partial fraction decomposition used?

Partial fraction decomposition is primarily used in integration, specifically with partial fraction integration. It can also be used to simplify complex rational expressions in algebra.

What is the process for partial fraction decomposition?

The process for partial fraction decomposition involves breaking down a rational function into its constituent parts, then setting up a system of equations to solve for the unknown coefficients. This requires knowledge of algebra and solving systems of equations.

What are the different types of partial fraction decomposition?

There are two main types of partial fraction decomposition: proper and improper. Proper decomposition involves breaking down a rational function into simpler fractions with proper denominators (i.e. no common factors between the numerator and denominator). Improper decomposition involves breaking down a rational function with improper fractions (i.e. the degree of the numerator is equal to or greater than the degree of the denominator).

What are the applications of partial fraction decomposition?

Partial fraction decomposition is commonly used in calculus, specifically in integration and solving differential equations. It can also be used in engineering and physics to solve for unknown variables in complex equations.

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