Given this PDE

$$\frac{1}{2}\frac{\partial^2 u}{\partial x^2}+(\sin(x)+\cos(x))\frac{\partial^2u}{\partial x\partial y}+(\sin(x)\cos(x))\frac{\partial^2u}{\partial y^2}=0 \quad \Omega= \mathbb{R}^2.$$

I was able to show that it is hyperbolic by filling the matrix with the coefficients \begin{bmatrix}\frac{1}{2} & \sin(x)+\cos(x) \\ \sin(x)+\cos(x) & \sin(x)\cos(x) \end{bmatrix} which has always a negativ determinant. At least I think that is correct.

My problem is I am not able to proof if and why there is a unique solution given $$u(0,y)=f(y) \text{ and } \frac{\partial u}{\partial x}(0,y)=g(y)$$

I think I need the method of characteristics which would show me that those lead to a unique solution. But I did not manage to do so.



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