Unable to solve coupled differential equations in Maple

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)])

I have tried using des but don't get the correct solution

des := {`&x`((diff(f(x), x))^2, g(t))-`&x`((diff(g(t), t))^2, f(x)) = 0, `&x`(diff(g(t), t, t), f(x))-`&x`(diff(f(x), x, x), g(t)) = 0};

dsolve(des);

Thus, I have still been unsuccesful. Any help in this regard would be appreciated. Thank you so much.