How to find the integrating factor? (1st order ODE)

In summary, the conversation discusses finding the integrating factor for the given equation and the use of different methods to solve the ode. It is noted that one cannot mix infinitesimals and non-infinitesimals in the equation. The integrating factor is found to be -1 / (x2 + y2).
  • #1
hellotheworld
4
0
x2 + y + y2dx - x dy = 0

Integrating factor, I(x,y) = -1 / (x2 + y2)

How to find the integrating factor ?Why I cannot use below method to solve the ode ?

(1/N)(My - Nx) = g(x) , I(x,y)=exp( ∫ g(x) dx) OR
(1/M)(My - Nx) = h(y) , I(x,y)=exp( -∫ h(y) dy)
 
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  • #2
hellotheworld said:
x2 + y + y2dx - x dy = 0
That equation makes no sense. You cannot mix infinitesimals and non-infinitesimals like that.
 
  • #3
hellotheworld said:
x2 + y + y2dx - x dy = 0
Did you mean ##(x^2 + y + y^2)dx - x dy = 0##?
hellotheworld said:
Integrating factor, I(x,y) = -1 / (x2 + y2)

How to find the integrating factor ?Why I cannot use below method to solve the ode ?

(1/N)(My - Nx) = g(x) , I(x,y)=exp( ∫ g(x) dx) OR
(1/M)(My - Nx) = h(y) , I(x,y)=exp( -∫ h(y) dy)
 

1. What is an integrating factor and why is it important in solving 1st order ODEs?

An integrating factor is a function that is used to simplify the process of solving a 1st order ordinary differential equation (ODE). It is important because it allows us to convert a difficult equation into a much simpler form that can be easily solved.

2. How do I determine the integrating factor for a given 1st order ODE?

The integrating factor for a 1st order ODE can be determined by using the formula e∫P(x)dx, where P(x) is the coefficient of the y' term in the ODE. Alternatively, you can also use the method of inspection to determine the integrating factor.

3. Can I use any function as an integrating factor for a 1st order ODE?

No, not all functions can be used as an integrating factor for a 1st order ODE. The function must satisfy certain conditions, such as being non-zero and having a derivative that can be easily integrated. In most cases, the integrating factor is a function of the independent variable, x.

4. Is there a specific order in which I should solve for the integrating factor and the solution of the 1st order ODE?

There is no specific order in which you should solve for the integrating factor and the solution of the 1st order ODE. However, it is recommended to first find the integrating factor and then use it to solve the ODE, as it simplifies the process and makes it easier to solve the equation.

5. Can I use the integrating factor method to solve higher order ODEs?

No, the integrating factor method is only applicable to 1st order ODEs. For higher order ODEs, other methods such as the method of undetermined coefficients or variation of parameters should be used.

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