Solve the equation xux yuy 0 with the auxiliary condition u

Solve the equation xu_x +yu_y = 0 with the auxiliary condition u(1,x) = x^2

Solution-

xux + yuy = 0.

By the method of characteristics,

P = x, Q = y , R = 0

1/y dy = 1/x dx and du = 0

log y = logx + c,

log (y/x) = c

or, y = xe^c

or, y/x = K = e^c.

u(x,y) = f(y/x) is general solution.

Application of the boundry conditions u=x^2 on the line y=x and x=1 now determines function f because on y=x
u (1,x) = x^2

f (x) = x^2 means f(t) = t^2

thus, u(x,y) = log (y/x)^2

u(x,y) = 2log (y/x) is the final answer.