Derive the Analytical Solution - Stuck: y'' = (4*K/R)*sinh(y/2) + (K^2)*sinh(y)

[BlastOff]

Derive the Analytical Solution - y'' = (4*K/R)*sinh(y/2) + (K^2)*sinh(y)

Need to find the analytical solution y(x) for the differential equation below.

y'' = (4*K/R)*sinh(y/2) + (K^2)*sinh(y)

Both y and y' are equal to zero at infinity
Both K and R are constants.

I can use the reduction of order technique to get through the first integral and obtain:

(y')^2 = (16*K/R)*cosh(y/2) + 2*(K^2)*cosh(y) - (16*K/R) - 2*(K^2)

After this, I've been told that the RHS of the equation can be manipulated into a product of squared terms. In order to do this, I've been told there should be an intermediate step,

(A/B + 1)*B

where B is a product of squared terms, and the expression inside the parentheses can itself be manipulated into a squared term.

This is where I've gotten myself stuck. I've tried to make some half-angle substitutions, use the exponential expressions for sinh and cosh, etc. Nothing has worked yet. Been at it for a month now, and could really use some help. Thanks in advance.

 

[HallsofIvy]

Wouldn't it be simplest just to integrate the original equation twice? Surely you know how to integrate "sinh(y/2)" and sin(y)!

 

[BlastOff]

Wouldn't it be simplest just to integrate the original equation twice? Surely you know how to integrate "sinh(y/2)" and sin(y)!

So you're saying to do this...?

dy/dx = Integral [ (4*K/R)*sinh(y/2) + (K^2)*sinh(y) ] dx

where the above would be the first integral? Don't see how that works... x is the independent variable here, with y being an unknown function of x. So how would you go about integrating sinh(y) wrt x? Please bear with me if I'm missing something, but I'm not following your explanation. Maybe you could post some of the steps? Thanks.

Would love to see an analytical solution if anyone manages to find it.