Can a Riccati-type Differential Equation with a Square Root Term be Classified?

[Elros]

Hi Folks..I came up with a weird looking DE for my research

Q''[t] = f[t]Q'[t] + g[t]Q[t] + h[t]*Sqrt[(Q[t])^2 + (k[t]Q'[t])^2]

The thing that mixes my mind is the square root term..If there is no square root terms it would like a riccati type..but there is..i tried to google it but it just showed me bunch of papers about fractional power DEs which is different...Is there any classification for this type ?

Thanks..

 

[Elros]

Common any ideas ?

 

[picard]

Actually, setting:

$$Q(t)=e^{\int_0^t dx G(x)}$$

reduces your 2nd order ODE to a 1st order nonlinear one:

$$G'(t)=-G(t)^2+f(t) G(t)+g(t)+h(t) \sqrt{G(t)^2 k(t)^2+1}$$

Depending on the form of the functions f(t),h(t), g(t) and k(t) you might be able to solve the latter and then again it might be worse...