First Order Non-Linear Ordinary D.E.

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Discussion Overview

The thread discusses the challenges of solving a first-order non-linear ordinary differential equation (D.E.) using various methods, including exact equations and substitutions. Participants explore different approaches and solutions while addressing potential errors in the problem statement.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in solving the D.E. using exact methods and substitution.
  • Another suggests using the substitution \(v=\frac{y}{x}\) and references a final answer, but there are concerns about typos in the D.E.
  • Participants identify missing parentheses in the original D.E., which may have led to confusion and incorrect solutions.
  • One participant notes that even advanced tools like Wolfram cannot solve the D.E. in closed form, suggesting numerical approximations may be possible.
  • Another participant mentions trying to use WolframMathematica but struggles with the syntax and understanding of the problem.
  • A later contribution states that \(y=x\) is a particular solution to the equation, providing at least one known solution.
  • One participant claims that Maple was able to solve the problem, providing a specific solution involving an arctangent function and suggesting a method to reorganize the D.E. for integration.

Areas of Agreement / Disagreement

Participants generally agree that there are issues with the original D.E. due to typos. However, there is no consensus on a definitive method for solving the equation, and multiple approaches and opinions remain contested.

Contextual Notes

There are unresolved issues regarding the correct formulation of the D.E., including missing parentheses and the implications of these errors on the solutions proposed. The discussion also highlights the limitations of certain computational tools in handling this type of D.E.

dearcomp
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Hello people,

I couldn't solve the given D.E by using exact d.e & substitution method :(

Thanks in advance.

(x*y*sqrt(x^2-y^2) + x)*y' = (y - x^2*(sqrt(x^2-y^2) )

gif file of d.e can be found in the attachments part.
 

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Re: First Order Non Linear Ordinary D.E.

dearcomp said:
Hello people,

I couldn't solve the given D.E by using exact d.e & substitution method :(

Thanks in advance.

(x*y*sqrt(x^2-y^2) + x)*y' = (y - x^2*(sqrt(x^2-y^2) )

gif file of d.e can be found in the attachments part.

Hi dearcomp, :)

Use the substitution, \(v=\frac{y}{x}\). The final answer is given >>here<<.
 
Re: First Order Non Linear Ordinary D.E.

dearcomp said:
Hello people,

I couldn't solve the given D.E by using exact d.e & substitution method :(

Thanks in advance.

(x*y*sqrt(x^2-y^2) + x)*y' = (y - x^2*(sqrt(x^2-y^2) )

gif file of d.e can be found in the attachments part.

Sudharaka said:
Hi dearcomp,

Use the substitution, \(v=\frac{y}{x}\). The final answer is given >>here<<.

Hi dearcomp, welcome to MHB! :)

Can you verify that you have given the proper DE?
There is at least 1 typo with a missing parenthesis.

In Sudharaka's solution there is another missing parenthesis, meaning a different DE was solved.
 
Re: First Order Non Linear Ordinary D.E.

Yes Sudharaka solved a different D.E

This is the exact problem with the corrected parantheses..

(x*y*sqrt(x^2-y^2) + x)*y' = (y - x^2*(sqrt(x^2-y^2)))

Thank you in advance
 
Re: First Order Non Linear Ordinary D.E.

dearcomp said:
Yes Sudharaka solved a different D.E

This is the exact problem with the corrected parantheses..

(x*y*sqrt(x^2-y^2) + x)*y' = (y - x^2*(sqrt(x^2-y^2)))

Thank you in advance

In that case I'm afraid I can't help you.
It looks like the type of DE that monster Wolfram would be able to solve if humanly possible.
But Wolfram can't, or at least not in closed form.
Of course a numerical approximation is still possible.

Do you have a context for the problem?
Are there perhaps hints, suggestions, or is there specific course material that relates to the DE?
 
Re: First Order Non Linear Ordinary D.E.

I like Serena said:
In that case I'm afraid I can't help you.
It looks like the type of DE that monster Wolfram would be able to solve if humanly possible.
But Wolfram can't, or at least not in closed form.
Of course a numerical approximation is still possible.

Do you have a context for the problem?
Are there perhaps hints, suggestions, or is there specific course material that relates to the DE?

Indeed, even wolfram can't handle this problem.
I'm trying to use WolframMathematica but I can not make it understand the question because I'm not familiar with it...

There are no hints, suggestions related to the question :(
 
Re: First Order Non Linear Ordinary D.E.

I like Serena said:
Hi dearcomp, welcome to MHB! :)

Can you verify that you have given the proper DE?
There is at least 1 typo with a missing parenthesis.

In Sudharaka's solution there is another missing parenthesis, meaning a different DE was solved.

dearcomp said:
Yes Sudharaka solved a different D.E

This is the exact problem with the corrected parantheses..

(x*y*sqrt(x^2-y^2) + x)*y' = (y - x^2*(sqrt(x^2-y^2)))

Thank you in advance

Yes indeed it has missing parenthesis and I typed it incorrectly in Wolfram. Sorry about that.

dearcomp said:
Indeed, even wolfram can't handle this problem.
I'm trying to use WolframMathematica but I can not make it understand the question because I'm not familiar with it...

There are no hints, suggestions related to the question :(

I am not sure how we can solve this in the general case, but notice that \(y=x\) is a particular solution of this equation. Now at least you have one solution. :p
 
Re: First Order Non Linear Ordinary D.E.

Maple solved the problem giving

$\arctan \left( {\dfrac {y }{\sqrt {{x}^{2}-
y ^{2}}}} \right) +\dfrac{1}{2}\left(y^2+{x}^{2}\right)=c_1$

The way I would go about trying to solve this is the following. Expand your ODE and regroup according to the $\sqrt{x^2-y^2}$ So

$x \sqrt{x^2-y^2} (y y' + x) + xy' - y = 0$

or

$y y' + x + \dfrac{xy'-y}{x \sqrt{x^2-y^2}} = 0$

$y y' + x + \dfrac{xy'-y}{x^2 \sqrt{1-\left(\dfrac{y}{x}\right)^2}} = 0$

$\dfrac{1}{2} \dfrac{d}{dx} \left( x^2+y^2 \right) + \dfrac{\dfrac{d}{dx} \left(\dfrac{y}{x}\right)}{ \sqrt{1-\left(\dfrac{y}{x}\right)^2}} = 0$.

Now each piece can be integrated separately.
 

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