I hope someone can help me solve this problem, it is supplementary problem 2.53d from Schaum's book: Schaum's Outline of Complex Variables, 2nd ed.
I'm sure $u(x,y)$ is harmonic because I checked $\nabla^2u(x,y) = 0$.
Also, the Cauchy–Riemann equations are:
$$\frac{\partial v}{\partial y} = \frac{\partial u}{\partial x}= −2ye^{−2xy}\sin(x^2−y^2)+2xe^{−2xy}\cos(x^2−y^2),$$
$$\frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y}= −2xe^{−2xy}\sin(x^2−y^2)-2ye^{−2xy}\cos(x^2−y^2).$$
But when I integrate in order to find $v(x,y)$, I bump into integrals I can't solve, I tried integrating by parts but it seemed to me that it didn't work.
$$v = \int (-2ye^{-2xy} \sin(x^2-y^2) +2xe^{-2xy} \cos(x^2-y^2))dy.$$