2nd Order Nonlinear ODE Question

[frank1234]

Homework Statement

Solve the 2nd order nonlinear differential equation, with initial conditions y(0)=0 and y'(0)=1

y''=2ay^3-(a+1)y with a within [0,1]

It would be greatly appreciated if someone could point me in the right direction on this. Thanks!

Homework Equations

The Attempt at a Solution

I have rearranged the equation to be y''+ya+y=2ay^3

 

[LCKurtz]

Homework Statement

Solve the 2nd order nonlinear differential equation, with initial conditions y(0)=0 and y'(0)=1

y''=2ay^3-(a+1)y with a within [0,1]

It would be greatly appreciated if someone could point me in the right direction on this. Thanks!

Homework Equations

The Attempt at a Solution

I have rearranged the equation to be y''+ya+y=2ay^3

Not sure how much help it is to you, but assuming $a>0$ Maple gives a solution in terms of Jacobi functions:$$
y = \textrm{JacobiSN}(x,\sqrt a)$$

 

[frank1234]

Hmmmm, I am not sure if that helps. I will have to give that some thought...

 

[frank1234]

Any other suggestions on how to go about solving this would be helpful.

 

[pasmith]

Use the identity $$y'' = y'\frac{dy'}{dy} = \frac12 \frac{d}{dy}(y'^2).$$