Analytically Solving a Differential Equation with Constants a, b, and c

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Discussion Overview

The discussion revolves around the analytical solution of the differential equation yy' - axy = bx^5 - cx^3, where a, b, and c are constants. Participants explore various methods for solving the equation, including the use of integrating factors, transformations, and power series, while expressing confusion and uncertainty about the approaches and the implications of boundary conditions.

Discussion Character

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

Main Points Raised

  • One participant requests help in solving the equation analytically, expressing confusion about the next steps.
  • Another suggests finding an integrating factor for the equation.
  • A transformation to t = x² is proposed, leading to a linear ordinary differential equation (ODE), though some participants question the simplification.
  • Concerns are raised about the equivalence of the transformed form to the original equation, with one participant doubting the transformation's effectiveness.
  • Power series solutions are mentioned as a potential method, though difficulties in determining the radius of convergence are noted due to the yy' term.
  • One participant expresses skepticism about deriving a closed form for the solution in general, suggesting it may only be possible under specific conditions for a, b, and c.
  • Boundary conditions y(0) = finit and y(-infinit) = finit are introduced, with questions about their sufficiency for deriving a closed form solution.
  • Another participant argues that only one boundary condition can typically be satisfied for a first-order ODE and questions the usability of the second condition in the context of a limited series development.
  • Concerns are raised about the physical relevance of the ODE given the boundary conditions, suggesting that the equation may not model the intended phenomena adequately.

Areas of Agreement / Disagreement

Participants express various viewpoints on the methods for solving the equation and the implications of the boundary conditions. There is no consensus on the best approach or the feasibility of obtaining a closed form solution, indicating multiple competing views remain.

Contextual Notes

Participants highlight limitations regarding the assumptions made in the transformation and the applicability of boundary conditions, particularly in relation to the behavior of the series solutions at large negative or positive values of x. The discussion reflects uncertainty about the physical interpretation of the boundary conditions in relation to the differential equation.

fatimajan
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Hello every body,
would you please help me to solve this equation analytically?
actually I'm confused and I don't know what to do?
yy'-axy=bx^5-cx^3
where a,b,c are constants
thank you
 
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Find an integrating factor for the equation.
 
Let t=x². This leads to a linear ODE
 
Thank you for your guidanc, jjacquelin.
do you mean yy'-at^(1/2)y=bt^(5/2)-ct^(3/2) ?
well I see no difference between this and the initial form!
maybe I'm wrong?
 
Don't confuse y'(x) and y'(t) . They aren't equal :
y'(x) = y'(t)*2*t^(1/2)
So the transformed form is simpler than the initial :
2yy'-ay = bt²-ct
But, it isn't a linear ODE. I admit my mistake.
 
fatimajan said:
Hello every body,
would you please help me to solve this equation analytically?
actually I'm confused and I don't know what to do?
yy'-axy=bx^5-cx^3
where a,b,c are constants
thank you

Ok, it looks like the equation in t is just as difficult and you asked for an analytic solution. You could solve it via power series to achieve that goal although I think it's difficult to determine the radius of convergence of a Cauchy product that would enter into the power series solutions due to the yy' term.
 
I agree with jackmell's comments :
Ok, it looks like the equation in t is just as difficult and you asked for an analytic solution. You could solve it via power series to achieve that goal although I think it's difficult to determine the radius of convergence of a Cauchy product that would enter into the power series solutions due to the yy' term.
The series development leads to complicated formulas for the coefficients as functions of a, b and c (Attachment below).
It is doubtfull that a closed form can be derived in the general case. May be possible in particular cases, for particular values of a, b, c, or in case of particular relationship between these parameters. This would require specific studies if more information is available concerning the parametrers values or relationship beteween them.
 

Attachments

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Thank you for your help, jackmell and jjacquelin

but dear jjacquelin,
actually I don't know about power series method well to solve the equation like you again, since I think you've made a mistake just in transformed form (yy' in lieu of 2yy') in your attachment, so I want to know this mistake makes what difference in your solution that I assumed is true?
and the last question, I have two boundary conditions:
y(0)=finit
y(-infinit)=finit

do you think they're enough to derive a closed form for our solution?

Thank you again
 
Hello,
I think you've made a mistake just in transformed form (yy' in lieu of 2yy') in your attachment
You are right, but it is just a typing mistake. I forgot the "2" in typing the equation, but I didn't forget the "2" in the calculs. So the series development is correct.
I have two boundary conditions:
y(0)=finit
y(-infinit)=finit
do you think they're enough to derive a closed form for our solution?
Since it is a first order ODE, only one boundary condition can be settled. If you set two conditions, generally a contradiction will occur.
The bondary condition y(0)=finit sets the yo value appearing in the formulas of the coefficients.
Moreover, a condition such y(-infinit)=finit isn't usable in case of limited series development. The series provides an approximate solution only for not too large values of abs(x), but not for x approaching -infinity or +infinity.
So, don't expect that the solution given in terms of a limited series will be satisfactory in case of large negative x values.

Note:
It seems that the ODE :
yy'-axy=bx^5-cx^3
associated with the boundary condition:
y(-infinit)=finit
has no solution. So, the ODE migth be not convenient to model the physical phenomena which gives y(-infinit)=finit
As a matter of fact, if y(-infinit) is finit, then y'(-infinity)=0
and cx^3 tends to be negigible compared to bx^5. So, the relationship tends to become equivalent to :
-axy=bx^5
With a finit value of y, this is impossible, because -axy isn't equivalent to bx^5, except if (a=0 and b=0), or if (y=0 and b=0).
 
Last edited:

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