First Order Non-Linear ODE (what method to use?)

In summary, the conversation discusses a problem of solving dy/dx = −[2x + ln(y)]*(y/x), with the attempt to check if it is exact, separable, homogeneous, or linear. The suggestion is to use the substitution y=et and solve by finding an integrating factor, which leads to a final solution of q = (A/x2) - 1.
  • #1
Apothem
39
0
Hi,

The problem is to solve:
dy/dx = −[2x + ln(y)]*(y/x)

Attempt:
I have tried to see if it is exact, I found it not to be, I can't easily find a function to multiply by to make it exact either (unless I am missing something obvious). It clearly isn't seperable, nor is it homogenous (I know although some non-homogenous ODEs can be solved using the homogenous method I don't think this one can be), it also isn't a linear equation so we can't use integrating factors.

I'm looking for pointers in the right direction/which method to use.

Thanks for your time
 
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  • #2
Have you tried guessing a solution?

Something like ##a^x## or something that works well with the natural log function.
 
  • #3
Did you figure it out?
 
  • #4
I think you are going the right way; checking the exact form, homogeneity etc.
Here since there is a logarithmic function, I recommend substituting y=et where t is parameter and then be solved by finding integrating factor.
 
  • #5
Using y = exp(qx), where q is a function of x , I got it to separate, giving
q = (A/x2) - 1, where A is a constant of integration.
It seems to check.
 

1. What is a first order non-linear ODE?

A first order non-linear ODE (ordinary differential equation) is a type of mathematical equation that involves a single independent variable and its derivatives. The equation is non-linear because the dependent variable and its derivatives are raised to powers or multiplied together.

2. How do I know which method to use for solving a first order non-linear ODE?

There are several methods that can be used to solve a first order non-linear ODE, including separation of variables, substitution, and integrating factors. The best method to use will depend on the specific form of the equation and the initial conditions given. It is important to have a good understanding of each method and to try different approaches to see which one yields the most accurate solution.

3. Can a computer program be used to solve first order non-linear ODEs?

Yes, there are many software programs and programming languages that have built-in functions for solving first order non-linear ODEs. These programs use numerical methods to approximate the solution to the equation. However, it is important to have a good understanding of the underlying math involved in order to properly use and interpret the results from these programs.

4. Are there any real-life applications of first order non-linear ODEs?

Yes, first order non-linear ODEs have many applications in science and engineering. They can be used to model physical systems such as population growth, chemical reactions, and electrical circuits. They are also used in fields such as economics, biology, and epidemiology to study and predict behavior of complex systems.

5. Is it possible to have multiple solutions to a first order non-linear ODE?

Yes, it is possible to have multiple solutions to a first order non-linear ODE. This can occur when the equation is not well-defined or when there are multiple initial conditions that satisfy the equation. It is important to carefully consider the problem and any given conditions in order to determine the appropriate solution.

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