How Do I Solve This Complex ODE Involving y''+C1*y^2+C2*y=0?

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Discussion Overview

The discussion revolves around solving the complex ordinary differential equation (ODE) given by y'' + C1*y^2 + C2*y = 0. Participants explore various approaches and insights related to this non-linear equation, including hints, methods, and potential solutions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in solving the ODE and seeks assistance.
  • Another participant provides a hint by suggesting the use of differentiation notation but later corrects themselves, noting that the presence of the y^2 term indicates the equation is non-linear.
  • A different participant proposes that the ODE can be transformed into an exact form using an integrating factor, leading to a first-order ODE that can be solved by separation of variables.
  • Another suggestion involves looking into elliptic functions and their associated differential equations as a potential avenue for solutions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on a single method for solving the ODE, and multiple competing views and approaches are presented throughout the discussion.

Contextual Notes

There are unresolved assumptions regarding the nature of the constants C1 and C2, and the specific conditions under which the proposed methods may be applicable. The discussion also highlights the complexity introduced by the non-linear term in the equation.

Who May Find This Useful

This discussion may be useful for individuals interested in advanced differential equations, particularly those involving non-linear terms, as well as those exploring various mathematical techniques for solving such equations.

hyime
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This is really difficult for me(at least for me), though it seems simple!
Could anyone help me to solve it or give some suggestion?

y''+C1*y^2+C2*y=0

Thank you!
 
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Welcome to PF!

Hi hyime! Welcome to PF! :smile:

(try using the X2 tag just above the Reply box :wink:)

Hint: if D represents "differentiate", then this is (D2 + C1D + C2)y = 0 :smile:
 


tiny-tim said:
Hi hyime! Welcome to PF! :smile:

(try using the X2 tag just above the Reply box :wink:)

Hint: if D represents "differentiate", then this is (D2 + C1D + C2)y = 0 :smile:

No, it isn't. The second term has [itex]y^2[/itex], not y'. That's a non-linear d.e. and is NOT simple.
 
HallsofIvy said:
No, it isn't. The second term has [itex]y^2[/itex], not y'. That's a non-linear d.e. and is NOT simple.

oops! :biggrin: …

show how it pays to write clearly! :smile:​
 
The integrating factor to your ODE is y', so

y'(y''+C1*y^2+C2*y)=0

is an exact ODE, that is it can be presented as

(2/3*C1*y^3+C2*y^2+(y')^2)'=0

or

2/3*C1*y^3+C2*y^2+(y')^2=c

where c is an arbitrary constant. The last first order ODE (with constant coefficients) is solvable by "separation of variables".
 
Look for elliptic functions and it's associated differential equation.
 
Thank you for all your help, I am working on it.
 

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