Solving ODE: Integrating Factor for Problem 4d

In summary, the conversation is about attempting to solve a specific ODE and finding an integrating factor, with a link to the problem set provided for clarification. The individual is also given some hints and suggestions on how to solve the equation.
  • #1
roryhand
2
0
Howdy, I've read this forum for some time, however this is my first post. I am attempting to solve this ODE. I am looking to find an integrating factor, then solve. I have attached the link to the problem set if my input here is ambiguous. Number 4d. Thank you kindly for any help you might lend.


(2x^2)+y+((x^2)*y)-x)dy/dx=0

My reasoning takes me as far as the integrating factor being exp(int( ? )dx)

https://people.creighton.edu/%7Elwn70714/DE_Assignments/DE%20Assignment%202%20PDF.pdf [Broken]
 
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  • #2
(2x^2)+y+((x^2)*y)-x)dy/dx=0

(2x^2+y)dx + [(x^2)*y-x]dy=0

Now you just need re-read your text-book about how to solve Pdx+Qdy=0, after checking some conditions on P&Q if such an equation has roots or none.
 
  • #3
Or can approach it this way:

[tex](2x^2+y)dx+(x^2y-x)dy=0[/tex]

So, after a quick check for homogeneous, exact, and explicit calc. of an integrating factor via partials, we expand the differentials and attempt to group them together to form exact differentials:

[tex]2x^2dx+ydx+x^2ydy-xdy=0[/tex]

Well, the ydx-xdy can be grouped as:

[tex]y^2\left(\frac{ydx-xdy}{y^2}\right)[/tex]

This leaves us with:

[tex]2x^2dx+x^2ydy+y^2 d\left(\frac{x}{y}\right)[/tex]

Can you re-arrange this now to obtain exact differentials which can be integrated?
 

1. What is an ODE?

An ODE, or ordinary differential equation, is a mathematical equation that describes the relationship between a dependent variable and its derivatives with respect to an independent variable.

2. What is an integrating factor?

An integrating factor is a function that is multiplied to both sides of an ODE in order to make it easier to solve. It is typically used for solving non-linear or non-separable ODEs.

3. How do you find the integrating factor for an ODE?

To find the integrating factor for an ODE, you must first identify the coefficient of the highest derivative term. Then, you can use the formula e∫p(x)dx, where p(x) is the coefficient, to find the integrating factor.

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

The purpose of using an integrating factor is to transform a non-linear or non-separable ODE into a linear or separable one, which can be easier to solve. It is also used to find a general solution to the ODE.

5. What is the process for solving an ODE using an integrating factor?

The process for solving an ODE using an integrating factor involves finding the integrating factor, multiplying it to both sides of the ODE, and then using integration to solve for the dependent variable. Finally, the general solution can be obtained by adding the constant of integration.

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