Suppose f is a non-constant differentiable function of one variable.
Determine, with reasons, the value of $\lambda$ for which
F(x, y) = f($\lambda x^{3}$ + y)
satisfies the partial differential equation
$\frac{\partial F}{\partial x}$+ $x^{2}\frac{\partial F}{\partial y}$= 0.
I tried to treat f($\lambda x^{3} + y)$ as a normal equation as $\lambda x^{3} + y$
and do partial differentiation
$\frac{\partial F}{\partial x}$ and $\frac{\partial F}{\partial y}$
$\frac{\partial F}{\partial x}$ = 3$\lambda x^{2}$
$\frac{\partial F}{\partial x}$ = 1
then I substitute these two answers back to the partial differential equation:
3$\lambda x^{2}$+$x^{2}$=0
$x^2$(3$\lambda$+1)=0
$\lambda$=$\frac{-1}{3}$
I am not sure if I am right or not, would someone help please?