Differentiating both sides of a DE

Question

In general if you have a differential equation with two variables such that: $$L(x,y)=h_1[f(x),f'(x),f^{(2)}(x),...,f^{(n)}(x),g(y),g'(y),g^{(2)}(y),...,g^{(n)}(y)]\\ R(x,y)=h_2[f(x),f'(x),f^{(2)}(x),...,f^{(n)}(x),g(y),g'(y),g^{(2)}(y),...,g^{(n)}(y)]\\ L(x,y)=R(x,y) $$ Does the following hold true? $$\frac{\partial^a}{\partial x^a}\frac{\partial^b}{\partial y^b}L(x,y)=\frac{\partial^a}{\partial x^a}\frac{\partial^b}{\partial y^b}R(x,y),\phantom{..........} \forall a\in\mathbb{Z},\forall b\in\mathbb{Z}$$