How I can solve this differential equation use implicit function theorem?

Question

I have to find the solution of the next differential equation:

$$x'(t)=\frac{-\frac{\partial F(t,x(t))}{\partial t}}{\frac{\partial F(t,x(t))}{\partial x}}$$

I need help I get all mixed up with this problem...

PD: you know $\frac{\partial F(t,x(t))}{\partial t}$ and $\frac{\partial F(t,x(t))}{\partial x}$, thanks!

Answer 1

The solution is the implicit function defined by $$ F(t,x(t))=F(t_0,x_0)=const. $$ in some interval around $t_0$ with values in a neighborhood of $x_0$.

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Note that for an IVP $x'=f(t,x)$, $x(t_0)=x_0$ the equivalent Picard integral (fixed-point) equation is $$ x(t)=x_0+\int_{t_0}^t f(s,x(s))\,ds. $$ Thus not much information is gained as long as the unknown function $x(t)$ also appears inside the integrand.