I am unable to solve the following differential equation for u(t). I right-click and select Solve DE but it gives me an incomprehensible solution. Kindly tell me how what to do:

diff(u(t)^2, t, t)+`&x`(`&x`(alpha, u(t)), diff(u(t), t))+`&x`(beta, u(t)^3) = 0

The same equation in "String" notation:

"2*diff(u(t),t)^2+2*u(t)*diff(diff(u(t),t),t)+alpha*u(t)*diff(u(t),t)+beta*u(t)^3 = 0"

The same equation in D notation:

I wish to solve the following equations simultaneously:

(diff(g(t), t, t))*f(x)-(diff(f(x), x, x))*g(t) = 0

and

(diff(f(x), x))^2*g(t)-(diff(g(t), t))^2*f(x) = 0

I searched the help files and did the following:

I have tried using solve but get an error

solve({`&x`(diff(g(t), t, t), f(x))-`&x`(diff(f(x), x, x), g(t)) = 0, `&x`((diff(f(x), x))^2, g(t))-`&x`((diff(g(t), t))^2, f(x)) = 0}, [f(x), g(t)])