How Do You Solve Non-linear Second Order Differential Equations?

[athrun200]

Homework Statement

Homework Equations

The Attempt at a Solution

I just want to ask how to obtain the solution?
What mathematical method involved?

 

[ehild]

Replace back the given function into the differential equation and see if it is really the solution.

ehild

 

[Saladsamurai]

What makes you think this is Non Linear?

 

[athrun200]

Replace back the given function into the differential equation and see if it is really the solution.

ehild

I just want to know, if I only have the first line, how can I get second line?
Just by try and error?

 

[athrun200]

What makes you think this is Non Linear?

I see there is a square there.

 

[ehild]

I just want to know, if I only have the first line, how can I get second line?
Just by try and error?

Is the second line solution of the differential equation?

ehild

 

[ideasrule]

I see there is a square there.

It's still linear because neither psi nor the second derivative of psi is being squared. However, the second line isn't the solution, and there's no solution in terms of elementary functions:

http://www.wolframalpha.com/input/?i=y%27%27%3Dx^2*y

 

[athrun200]

It's still linear because neither psi nor the second derivative of psi is being squared. However, the second line isn't the solution, and there's no solution in terms of elementary functions:

http://www.wolframalpha.com/input/?i=y%27%27%3Dx^2*y

I saw the eqtn from this book.
Does it mean that the approximation is wrong?

 

[ehild]

If you calculate the second derivative of Ψ=exp(±z²/2) you will find that it is (z²±1)Ψ. Therefore the Ψ-s are only approximate solution, valid for z²>>1.

ehild