$u=u(x,y), \frac{{\partial}^2u}{\partial x^2}=\frac{{\partial}^2u}{\partial y^2}\Rightarrow u''_{11}=u''_{22}$?, where $u_{11}''$ is notation of second partial derivative with respect to the first variable, the same to $u_{22}''$
This question comes from the following problem.
If $$u'_1(x,2x)=x^2\;(1)$$ and $$u(x,2x)=x\; (2)$$, u's second order partial derivative is continuous and $\frac{{\partial}^2u}{\partial x^2}=\frac{{\partial}^2u}{\partial y^2}$,then what's $u''_{11}(x,2x)=$?
What the author did:
Take derivative of equation $(1)$ with respect to $x$, then take derivative of equation$(2)$ with respect to x(say, give us $(3)$), then take derivative of equation $(3)$ with respect to $x$. And the author use the property$u''_{11}=u''_{22}$, which seems unreasonable to me when $y$ is a function of $x$.