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

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In summary, the conversation is about solving a difficult and seemingly simple problem involving a non-linear differential equation. The participants are discussing possible methods and providing hints for solving the problem. The equation is eventually shown to be solvable by separation of variables and the use of elliptic functions.
  • #1
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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  • #2
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:
 
  • #3


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.
 
  • #4
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:​
 
  • #5
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".
 
  • #6
Look for elliptic functions and it's associated differential equation.
 
  • #7
Thank you for all your help, I am working on it.
 

1. How can I identify a difficult ODE?

A difficult ODE can be identified by its complexity, nonlinearity, and possible lack of analytical solutions. It may also involve higher-order derivatives or multiple variables.

2. What are some strategies for solving a difficult ODE?

Some strategies for solving difficult ODEs include using numerical methods, such as Euler's method or Runge-Kutta methods, or transforming the ODE into a simpler form through substitution or change of variables.

3. Can I use software to solve a difficult ODE?

Yes, there are many software programs and packages available that can solve difficult ODEs numerically. Some commonly used ones include MATLAB, Mathematica, and Python's SciPy library.

4. Are there any tips for tackling a difficult ODE?

Yes, some tips for tackling a difficult ODE include breaking it down into smaller, more manageable parts, using algebraic manipulations to simplify the equation, and considering boundary conditions or initial conditions to help guide the solution process.

5. What resources are available for getting help with a difficult ODE?

There are many online resources available, such as forums, online communities, and video tutorials, where you can ask for help with a difficult ODE. You can also consult with a mathematics or physics tutor or seek guidance from your peers or professors.

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