Using an integrating factor properly

In summary, The conversation is about trying to solve an equation with an integrating factor e^(-2t) and getting stuck on how to integrate the equation. The book simplifies the equation to d(e^(-2y))/dt=4e^(-2t)-te^(-2t) and the participants are discussing how this simplification occurs and how to use the product rule to solve it.
  • #1
cameuth
17
0
alright guys, I've been trying to tackle this for a couple of hours now.

dy/dt-2y=4-t
my integrating factor is e^(-2t) of course.

dy(e^(-2t))/dt-2ye^(-2t)=4e^(-2t)-te^(-2t)

then I get completely lost. how do I integrate when it's like this? My book simplifies the above equation into

d(e^(-2y))/dt=4e^(-2t)-te^(-2t)

can anyone explain how that simplification occurs??
 
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  • #2
hi cameuth! :smile:

(try using the X2 button just above the Reply box :wink:)
cameuth said:
d(e^(-2y))/dt=4e^(-2t)-te^(-2t)

can anyone explain how that simplification occurs??

(you mean d(ye-2t)/dt :wink:)

use the product rule on ye-2t :smile:​
 

1. What is an integrating factor?

An integrating factor is a mathematical function used to solve differential equations. It is multiplied with both sides of the equation to make it easier to integrate.

2. When should I use an integrating factor?

An integrating factor should be used when solving differential equations that are not in the form of separable variables or cannot be solved using other methods such as substitution or partial fractions.

3. How do I choose the right integrating factor?

The integrating factor is usually chosen based on the type of differential equation being solved. For linear first-order equations, the integrating factor is usually the exponential of the integral of the coefficient of the variable being integrated. For other types of equations, a common approach is to use the method of variation of parameters.

4. Can an integrating factor be used for higher-order differential equations?

Yes, an integrating factor can be used for higher-order differential equations. However, the process becomes more complicated as the order of the equation increases, and it may not always be possible to find an integrating factor that works for all solutions.

5. Are there any limitations to using an integrating factor?

Yes, there are some limitations to using an integrating factor. It may not work for all types of differential equations, and the process can become more complex for higher-order equations. Additionally, the choice of integrating factor may not always be obvious and may require some trial and error.

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