Understanding Integrating Factor for Linear ODEs

In summary, the use of integrating factor when solving first order linear differential equations is a technique that works for a sub-class of equations. Due to the subtle and complex nature of differential equations, there is limited general understanding and many consider it a bag of tricks. The rationale behind this technique is based on the product rule and can be derived through observation. Understanding the "why" behind this technique may be difficult to find in certain resources, but it is important for a deeper understanding of differential equations.
  • #1
erik006
5
0
Hi,

I'm learning differential equations, and although I understand the methods I have learned thus far, I often have trouble seeing what is the reasoning behind them.

Take for example, the use of the integrating factor when solving first order linear ODE's. I understand how to use it, but I'm not sure where it came from. In the resources that I'm using there's really not explanation, instead the discussion is limited to: assume there's a function by which we can multiply our differential equation to make it integrable.

Can anybody explain why we use this approach?

thanks,

Erik
 
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  • #2
Diff.eqs are subtle, complex and extremely varied creatures.
What this entails is that it is exceedingly difficult to generate theorems and general truths about them that at the same time is readily applicable.

Following this, is that whenever we DO have some powerful, easy technique that can be used on a sub-class of diff.eqs, the rationale behind that technique might be little else than "it works".

That is, our equipment for handling diff.eqs might be described as just a bag of tricks; isolated techniques that work only in a few special cases.

That is one of the reasons why many professional mathematicians shy away from working directly with the diff.eqs themselves, because it is difficult to find some interesting, general results there. The work they DO make, might well have an impact on generating new techniques for handling diff.eqs, as a side benefit.
 
  • #3
mathworld has a detailed derivation/proof/what are youmacallit.

just google first order linear differential equation.
 
  • #4
It's really based on the simple observation that d(p(t)y)dt=p(t)dy/dt+ p'(t)y. (The product rule.)

If you have a differential equation of the form dy/dt+ g(t)y= f(t), and you multiply by any function p(t), you have p(t)dy/dt+ p(t)g(t)y= p(t)q(t). Now compare the two forms. It should be obvious that the left side of your new equation is "exact" ( equals d(p(t)y)/dt) if and on p'(t)= p(t)f(t). That's separable equation, easy to solve for p(t).
 
  • #5
Thanks everyone for your responses! It's very much appreciated.

HallsofIvy, that's pretty much what I was looking to hear. The book I'm using my studies is "advanced engineering mathematics" by zill & cullen. Personally I think it's not worth reading, having or even using as a coaster. I looked at some other books, and one of them gave me a similar explanation to what you gave me.

I feel the need to understand the "why" behind ideas like that. It is not satisfying and somehow an obstruction to understanding to just know that it works. I suppose that's what you get in current engineering curricula...

Thanks again,

Erik
 

What is an integrating factor for linear ODEs?

An integrating factor for linear ODEs is a function that is multiplied to both sides of an ordinary differential equation (ODE) in order to simplify its solution. It is usually used when solving first-order linear ODEs, where the integrating factor helps to convert the equation into a form that is easier to integrate.

How do you determine the integrating factor for a linear ODE?

The integrating factor for a linear ODE can be determined by using the formula e∫P(x)dx, where P(x) is the coefficient of the linear term in the ODE. Once this factor is determined, it is then multiplied to both sides of the ODE to simplify the solution process.

What is the purpose of using an integrating factor in solving linear ODEs?

The main purpose of using an integrating factor in solving linear ODEs is to simplify the process of finding the solution. By multiplying both sides of the ODE with an integrating factor, the equation can be transformed into a form that is easier to integrate, making it more manageable to solve.

Can an integrating factor be used to solve nonlinear ODEs?

No, an integrating factor is only used to solve linear ODEs. Nonlinear ODEs require different methods and techniques for solving, and using an integrating factor will not work in these cases.

Are there any limitations to using an integrating factor in solving linear ODEs?

Yes, there are some limitations to using an integrating factor. It can only be used for first-order linear ODEs, and it may not always lead to a closed-form solution. In some cases, it may also introduce new solutions that are not valid for the original ODE.

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