<s>[INST] Find the orthogonal trajectories of the following family of curves: $x^3y - xy^3 = \alpha$ The answer is $-6x^2y^2 + x^4 + y^4 = \beta$, but I get $-3x^2y^2 + \frac{x^4}{4} + \frac{y^4}{4} = \beta$ and I can't seem to find my mistake. My work:
Differentiaing the given equation, we get 
$$3x^2y + x^3\frac{dy}{dx} - y^3 - 3y^2x\frac{dy}{dx} = 0. $$
So $$3x^2y - y^3 = (3y^2x -x^3)\frac{dy}{dx}, $$ and hence 
$$ \frac{dy}{dx} = \frac{3x^2y - y^3}{3y^2x - x^3}. $$
Now, the slope of the other family of curves will be the negative inverse, so for the orthogonal trajectory, we have $$\frac{dy}{dx} = -\frac{3y^2x-x^3}{3x^2y-y^3}, $$ which we can write as 
$$-(3y^2x-x^3)dx = (3x^2y-y^3)dy. $$ Integration of the last differential equation yields 
$$-\frac{3}{2}y^2x^2 + \frac{x^4}{4} = \frac{3}{2}x^2y^2 - \frac{y^4}{4} + \beta, $$ $\beta$ being an arbitrary constant. 
This simplifies to $$-3y^2x^2 + \frac{x^4}{4} + \frac{y^4}{4} = \beta. $$ which is not correct.
Where am I going wrong?
 [/INST] Since this is tagged complex analysis, 
let $z = x+iy$ and let $f(x+iy) = x^3y - xy^3$.
Expressed in terms of $z$, this is $f(z) = \frac 1 {16i}\left((z+\bar{z})^3(z - \bar z) + (z + \bar z)(z - \bar z)^3\right) = \frac 1{8i}(z^4 - \bar z^4) = \frac 1 4 Im(z^4) = Im(z^4/4)$.
So the curves $x^3y-xy^3 = A$ are the pullback of the horizontal lines by the holomorphic function $z \mapsto z^4/4$. Since holomorphic functions preserve angles (except when the derivative vanish, here it is at $z=0$), the curves that are always orthogonal to those, are the pullback of the vertical lines by $z \mapsto z^4/4$, i.e. the curves $Re(z^4/4) = B$.
And finally, $Re(z^4/4) = (x^4-6x^2y^2+y^4)/4$
At $z=0$, $f$ multiplies angles by $4$ and is $4$-to $1$ around $z=0$, so the two curves $Re(f(z)) = 0$ and $Im(f(z)) = 0$ both have a star-like singular point (an $8$-branched star), and the angle between the two stars is $\pi/8$ 
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