Diagonal of a two variables function and its partial derivative

Question

$L(x,y)$ is a nice function (we can assume nice properties of it if needed), now suppose $$\frac{\partial L(x,y)}{\partial y}|_{y=x}\equiv H(x)$$ is a known function, then what can we learn about the diagonal function $L(x,x)$?

Thanks a lot.

Answer 1

$$ \frac{\partial }{\partial x } L(x,x) =L_x(x,x)+L_y(x,x)=L_x(x,x) + H(x)$$ so that$$ L(t,t)-L(0,0)=\int_0^t (L_x(s,s)+ H(s))\ ds $$