Second order nonlinear differential equation

[equation M]

Is it possible to solve the following differential equation analytically?

y''(x) = A - B [exp(y(x)/C) - 1]

where A, B and C are constants.


Thank you...

 

[HallsofIvy]

that can be simplified to
y'= A- Be^{y/C}- B= (A- B)- Be^{y/C}= A'- Be^{y/C}
with A- B= A'
and if you let z= y/C, y= Cz, so y''= Cz'' and the equation becomes
Cz''= A'- Be^z or z''= A"- B'e^z
with A"= A'/C and B'= B/C.

Since the independent variable, x, does not appear in that, let u= z' so that z''= u'= (du/dz)(dz/dx)= u(du/dz) and the equation becomes
u(du/dz)= A"- B'e^z so udu= (A"- B'e^z)dz and, integrating,
(1/2)u^2= A"z- B'e^z+ C

So u= dz/dt= sqrt(2(A"z- B'e^z+ C))

dt= dz/sqrt(2(A"z- B'e^z+ C))

That looks like it cannot be integrated analytically but at least it is set up for a direct numerical integration.