Reducing Order: Is it Valid to Say G(x) = 0?

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In summary, the conversation discusses using reduction of order to solve differential equations. The standard form for this method is y"+ P(x)y' +G(x)y = 0, where P(x) and G(x) are continuous on a given interval. The question posed is whether the given DE y" - xy' = 0 with y=e^x and I= (0,infinity] can be considered under standard form with G(x) = 0. The answer is yes, and the conversation also mentions the importance of verifying solutions.
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bishy
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I'm making an assumption while trying to solve DEs by reduction of order. I've got a short form equation that I can use to reduce it if and only if I can place the DE into standard form. The standard form with generic notation would be y"+ P(x)y' +G(x)y = 0 where P(x) and G(x) are continuous and on the interval I. I am not sure if I am able to say the following, therefore my question would be is this valid:

Given the DE y" - xy' = 0; y=e^x; I= (0,infinity] is it valid to state that the DE is under standard form where G(x) = 0?
 
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Yes, of course it is! I am wondering what you mean by "y= e^x" immediately after the differential equation. That is clearly not a solution to the equation.
 
  • #3
ah ok. that's what I thought, I just wasn't sure. As for it not being a solution, I made it up and never bothered checking it. It's not a big a deal, most of the work I have done is confirmed with this.
 

What is the concept of "reducing order" in mathematics?

"Reducing order" refers to the process of simplifying an expression or equation by eliminating terms with a certain variable. This is commonly done by setting the expression or equation equal to zero and solving for the variable.

How do you determine if an expression or equation has been reduced to its lowest order?

If an expression or equation contains only the necessary terms and variables, and cannot be further simplified, it can be considered to be in its lowest order. In other words, all non-essential terms and variables have been eliminated.

Is it always valid to say that G(x) = 0 when reducing order?

No, this is not always the case. While setting an expression or equation equal to zero is a common method of reducing order, it may not always be valid. It depends on the specific context and variables involved in the expression or equation.

What are some situations where reducing order may not be valid?

Reducing order may not be valid in cases where the expression or equation involves certain mathematical operations or variables that do not allow for setting the expression or equation equal to zero. Additionally, in some cases, reducing order may not accurately represent the underlying mathematical relationship being studied.

What are some potential drawbacks of reducing order in mathematical analysis?

Reducing order can lead to oversimplification and loss of important information in some cases. It is important to carefully consider the context and variables involved before deciding to reduce order, as it may not accurately represent the complexity of the underlying mathematical relationship.

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