First Order Non-Linear Ordinary D.E.

In summary, Maple solved the problem giving$\arctan \left( {\dfrac {y }{\sqrt {{x}^{2}- y ^{2}}}} \right) +\dfrac{1}{2}\left(y^2+{x}^{2}\right)=c_1$
  • #1
dearcomp
7
0
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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  • #2
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<<.
 
  • #3
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.
 
  • #4
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
 
  • #5
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?
 
  • #6
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 :(
 
  • #7
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
 
  • #8
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.
 

FAQ: First Order Non-Linear Ordinary D.E.

1. What is a First Order Non-Linear Ordinary Differential Equation (D.E)?

A First Order Non-Linear Ordinary Differential Equation (D.E) is a type of mathematical equation that involves one independent variable, one dependent variable, and their derivatives. It is non-linear because the dependent variable is raised to a power other than 1, and it is ordinary because it does not involve partial derivatives.

2. How is a First Order Non-Linear Ordinary D.E. different from a First Order Linear Ordinary D.E.?

The main difference between a First Order Non-Linear Ordinary D.E. and a First Order Linear Ordinary D.E. is that the former involves the dependent variable raised to a power other than 1, while the latter has a linear relationship between the dependent variable and its derivatives. This makes non-linear equations more difficult to solve and often requires numerical methods.

3. What are some real-world applications of First Order Non-Linear Ordinary D.E.?

First Order Non-Linear Ordinary D.E. can be used to model a variety of physical phenomena such as population growth, chemical reactions, and fluid flow. They are also commonly used in economics, engineering, and biology to describe various systems and processes.

4. How can I solve a First Order Non-Linear Ordinary D.E.?

There are several methods for solving First Order Non-Linear Ordinary D.E., including separation of variables, substitution, and numerical methods such as Euler's method and Runge-Kutta methods. The choice of method depends on the specific equation and initial conditions given.

5. What are the limitations of First Order Non-Linear Ordinary D.E.?

First Order Non-Linear Ordinary D.E. are limited in their ability to accurately model systems with complex dynamics. They also may not have analytical solutions and require numerical methods for approximation. In addition, the initial conditions must be known precisely in order to obtain an accurate solution.

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