# Is -t the Correct Integrating Factor for this Diff Eq?

• jesuslovesu
In summary, the conversation is about solving a differential equation using an integrating factor. The equation in question is g' - g/t = t e^t, and there is confusion about the correct integrating factor. The individual has found that the integrating factor is typically 1/t, but in this case, it seems to be -t. They question whether the equation is in the correct form to find the integrating factor. The solution is found to be e^{-\ln t}=e^{\ln t^{-1}}=\frac{1}{t}.
jesuslovesu
[SOLVED] Integrating Factor Diff Eqs

## Homework Statement

g ' - g/t = t e^t

I'm trying to solve this, but I seem to have run into a problem, according to my book the integrating factor is 1/t, however I believe that it is -t

## The Attempt at a Solution

e^int(-1/t dt) = e^(-lnt) = -t
That is how I found the solution to all the problems requiring an integrating factor before however this situation seems to be different... Do I have the equation in the right form to find the integrating factor?

$$e^{-\ln t}=e^{\ln t^{-1}}=\frac{1}{t}$$

## 1. What is an integrating factor in differential equations?

An integrating factor is a function that is used to solve certain types of differential equations, specifically those that are not exact. It is multiplied to both sides of the equation to make it exact, allowing for a simpler solution.

## 2. How do you find the integrating factor for a differential equation?

To find the integrating factor, you must first identify the type of differential equation you are working with. Then, use a specific formula or method to find the integrating factor. For example, for a first-order linear differential equation, the integrating factor is e^(integral of P(x) dx), where P(x) is the coefficient of y.

## 3. Why is it important to use an integrating factor in solving differential equations?

Integrating factors are crucial in solving certain types of differential equations, as they allow us to transform the equation into a more manageable form. This makes it easier to find a solution and helps us avoid more complex methods of solving the equation.

## 4. Can an integrating factor be used for all types of differential equations?

No, integrating factors can only be used for certain types of differential equations, such as first-order linear equations or those that are not exact. For other types of equations, different methods may be required to solve them.

## 5. Are there any limitations to using an integrating factor in solving differential equations?

While integrating factors can greatly simplify the process of solving some differential equations, they may not work for all cases. In some situations, the integrating factor may not be able to make the equation exact, or it may introduce additional complexities. It is important to understand the limitations and when to use alternative methods of solving differential equations.

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