<s>[INST] Need help with level 3 Calculus problem 
Two curves $$C_1: ([f(y)]^{2/3} + ([f(x)]^{1/3}) = 0\quad\mbox{and}\quad C_2: [f(y)]^{2/3}+ [f(x)]^{2/3} = 12, $$ satisfying the relation $$ (x-y)f(x+y)-(x+y)f(x-y) = 4xy(x^2-y^2).$$
1.) Evaluate the area bounded by $C_1$ and $C_2$.
2.) Evaluate the area bounded by $C_2$ and $|x|+|y|=\sqrt{12}$.
3.) Evaluate the area bounded by $C_1$ and $x+y+2=0$.

It was hard to write math here so:
Have a look at this pic for clear question
 [/INST] According to Functional Equation : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x).
the general solution of the function equation 
$$(x-y)f(x+y)-(x+y)f(x-y) = 4xy(x^2-y^2)$$
is $f(x)=x^3+kx$ with $k\in\mathbb{R}$. So the problem seems to be ambiguous because $f$ (and therefore also the curves $C_1$ and $C_2$) is not uniquely determined.
Note that if we let $k=0$ then $f(x)=x^3$ and
$$C_1: \; y^2+x=0\quad \mbox{and}\quad C_2:x^2+y^2=12.$$
In this case the area bounded by $C_1$ and $C_2$ is
$$2\int_{-\sqrt{12}}^{-3}\sqrt{12-x^2}dx+2\int_{-3}^0\sqrt{-x}dx=
2\pi+\sqrt{3}.$$
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