Integrating Factor: -g(m/g) and its Derivative

In summary, when finding the derivative of -g, it becomes -g(m/g) instead of just (-g^2/2). This is because the factor of (m/g) comes from integrating the exponential and g is constant with respect to the variable t. However, if integrating with respect to g, the result would be -g^2/2 since everything else is constant. Additionally, the integral of g would also have to be found, which is gt.
  • #1
Ry122
565
2
In the following problem how does -g end up becoming -g(m/g)?
Isn't the derivative of -g just (-g^2/2)?
http://users.on.net/~rohanlal/integfact.jpg
 
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  • #2
I'm having trouble reading the scan, but from what I can tell, the factor of (m/c?) came from integrating the exponential, and g is constant with respect to the variable t. Then again, I don't have any context either.
 
  • #3
Yes, in this case g is a constant because you are integrating with respect to t. If you were integrating with respect to g (no clue why you would because I strongly suspect that this is the gravitational constant) then everything else would be a constant and you would get -g^2/2.
 
  • #4
Shouldn't I also have to find the integral of g, which would be gt?
 

1. What is an integrating factor?

An integrating factor is a function that is used to simplify the process of solving a differential equation. It is multiplied to both sides of the equation in order to make the left side of the equation a total derivative.

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

The integrating factor can be found by multiplying the equation by an appropriate function and then solving for that function. The function is typically found by using a specific formula or by using a known method for solving differential equations.

3. What is the purpose of using an integrating factor?

The purpose of using an integrating factor is to transform a differential equation into a simpler form that can be easily solved. It helps to eliminate variables and make the equation more manageable.

4. What is the relationship between the integrating factor and its derivative?

The integrating factor and its derivative are related through the equation -g(m/g), where g is the integrating factor and m is the derivative of the integrating factor. This relationship is used to simplify the process of solving a differential equation.

5. When should an integrating factor be used?

An integrating factor should be used when solving a differential equation that cannot be solved by other methods, such as separation of variables or substitution. It is particularly useful for solving linear differential equations with variable coefficients.

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