11.5.3.8 With variable coefficients

Problem: Find the solution of the second order linear equation, satisfying the given initial conditions (the Cauchy problem): \[ \begin{array}{l} u_{x x}+2 \sin x \cdot u_{x y}-\cos ^{2} x \cdot u_{y y}+u_{x}+ \\ +(\sin x+\cos x+1) u_{y}=0, \\ \left.u(x, y)\right|_{y=-\cos x}=1+2 \sin x, \\ \left.u_{y}(x, y)\right|_{y=-\cos x}=\sin x . \end{array} \]