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

[hyime]

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!

 

[tiny-tim]

Welcome to PF!

Hi hyime! Welcome to PF!

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

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

 

[HallsofIvy]

Hi hyime! Welcome to PF!

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

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

No, it isn't. The second term has $y^2$, not y'. That's a non-linear d.e. and is NOT simple.

 

[tiny-tim]

No, it isn't. The second term has $y^2$, not y'. That's a non-linear d.e. and is NOT simple.

oops! …

show how it pays to write clearly! ​

 

[kosovtsov]

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".

 

[gato_]

Look for elliptic functions and it's associated differential equation.

 

[hyime]

Thank you for all your help, I am working on it.