# A non-exact nonlinear first ODE to solve

• Nipuna Weerasekara
In summary, an ODE is considered nonlinear if it contains terms involving y, such as y^2 or sin(y). The difference between an exact and non-exact ODE is that an exact ODE can be solved using an integrating factor, while a non-exact ODE cannot. Not all nonlinear first order ODEs can be solved analytically and may require numerical methods. Some methods to solve a non-exact nonlinear first order ODE include Euler's method, Runge-Kutta method, or the shooting method. Non-exact nonlinear first order ODEs have real-world applications, such as modeling population growth, chemical reactions, and electrical circuits.
Nipuna Weerasekara

## Homework Statement

Solve the following equation.

## Homework Equations

( 3x2y4 + 2xy ) dx + ( 2x3y3 - x2 ) dy = 0

## The Attempt at a Solution

M = ( 3xy4 + 2xy )
N = ( 2x3y3 - x2 )

∂M/∂y = 12x2y3 + 2x
∂N/∂x = 6x2y3 - 2x

Then this equation looks like that the integrating factor is (xM-yN).
IF = x3y4 + 3x2y

then the new equation would be:

(3xy^3+2)/(x^2y^3+3x)dx + (2xy^3-1)/(xy^4+3y)dy = 0

but then if we find that this new equation is exact or not, it proves that this is not exact.
If so what should I do?

Note:
The given answer is:
x3y2 + x2/y = c

Nipuna Weerasekara said:

## Homework Statement

Solve the following equation.

## Homework Equations

( 3x2y4 + 2xy ) dx + ( 2x3y3 - x2 ) dy = 0

## The Attempt at a Solution

M = ( 3xy4 + 2xy )
The above should be ##M = 3x^2y^4 + 2xy##
Nipuna Weerasekara said:
N = ( 2x3y3 - x2 )

∂M/∂y = 12x2y3 + 2x
∂N/∂x = 6x2y3 - 2x
You have a mistake above.
If ##M = 3x^2y^4 + 2xy## then ##M_y = 12x^2y^3 + 2x##
Since My ≠ Nx, that's enough to show that the equation is not exact.

Nipuna Weerasekara said:
Then this equation looks like that the integrating factor is (xM-yN).
IF = x3y4 + 3x2y
I taught diff. equations a number of times, but this isn't a trick that I remember. Where does xM - yN come from?
Nipuna Weerasekara said:
then the new equation would be:

(3xy^3+2)/(x^2y^3+3x)dx + (2xy^3-1)/(xy^4+3y)dy = 0
I'm not following this at all. If you can find an integrating factor, you multiply both sides of the original diff. equation by it. If it really is an integrating factor, then the new equation will be exact, and you can get the solution.
Nipuna Weerasekara said:
but then if we find that this new equation is exact or not, it proves that this is not exact.
If so what should I do?

Note:
The given answer is:
x3y2 + x2/y = c

LCKurtz said:
I think the OP is referring, incorrectly, to the integrating factor method for non-exact equations.
I understand that he/she was trying to come up with an integrating factor. The link below presents the usual technique to start with when you're faced with an inexact equation: seeing if a function of x (##\mu(x)##) in the PDF below, and proceeding to a function of y alone (##\nu(y)##) if the previous try didn't work. What threw me was the immediate jump to xM - yN, which as you point out, is not the integrating factor.
LCKurtz said:
I haven't worked through this problem, but the method is described in the link http://users.math.msu.edu/users/sen/math_235/lectures/lec_5.pdf on pages 3 - 5.

There was a misinterpretation in the question by me... I solved it by myself. Thanks for the concern.

## 1. How do I determine if an ODE is nonlinear?

An ODE is considered nonlinear if it cannot be written in the form of y' = f(x,y). In other words, if it contains terms involving y, such as y^2 or sin(y), it is nonlinear.

## 2. What is the difference between an exact and non-exact ODE?

An exact ODE can be solved using an integrating factor, while a non-exact ODE cannot. This means that the solution of an exact ODE can be written in a closed form, while a non-exact ODE may require numerical methods to solve.

## 3. Can all nonlinear first order ODEs be solved analytically?

No, not all nonlinear first order ODEs can be solved analytically. Some may require numerical methods or approximations to find a solution.

## 4. How can I solve a non-exact nonlinear first order ODE?

There are various numerical methods that can be used to solve a non-exact nonlinear first order ODE, such as Euler's method, Runge-Kutta method, or the shooting method. These methods involve breaking the ODE into smaller steps and approximating the solution.

## 5. Are there any real-world applications of non-exact nonlinear first order ODEs?

Yes, many real-world systems can be modeled using non-exact nonlinear first order ODEs, such as population growth, chemical reactions, and electrical circuits. These ODEs can provide insights into the behavior of these systems and be used to make predictions or solve problems.

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