You can look through the simplification of second-order linear partial differential equations in textbooks.

According to the title, A 1 1 = 1, A 12 = 1, A22 =-3.

The value of the characteristic line dy/dx satisfies the solution of the characteristic equation a11* x2-2 * a12 * x+a22 = 0, that is

Dy/dx=3 or dy/dx=- 1, that is, d(3x-y)=0 or d(x+y)=0.

Variable substitution: Let \xi=3*x-y and \η= x+y, then the universal equation can be transformed into:

(\ Partial 2 University)/(\ Part \ Xi \ Part \ ETA) = 0

So the general solution of universal equation is u(\xi, \η)= f(\ Xi)+g(\η), where f and g are arbitrary derivative functions.

So u(x, y)=f(3*x-y)+g(x+y), and the specific forms of f and g can be determined by bringing in boundary conditions.

The result is f (x) = (sin (x/3)-x/3)/2 and g (x) = (x+sinx)/2.

Therefore, the solution of the definite solution problem is:

u(x，y)=(sin((3 * x-y)/3)-(3 * x-y)/3)/2+(x+y+sin(x+y))/2，-\ infty & lt； x & lt+\infty，y & gt0.