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evinda

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I want to solve the following problem:

$$u_x(x,y)+(x+y)u_y(x,y)=0 , x+y>1 \\ u(x,1-x)=f(x), x \in \mathbb{R}$$

How could I do it? Could we apply the method of characteristics? In my lecture notes, there is an example on which this method is applied.

This example is of the form $a(t,x,u) u_x+ b(t,x,u)u_t=c(t,x,u)$.

$$x_t(x,t)-u_x(x,t)=0, x \in \mathbb{R}, t>0 \\ u(x,0)=f(x), x \in \mathbb{R}$$Does it make a difference if the variable is $t$ or $y$ ?

Also, at the beginning, they took: $(x(0),t(0))=(x_0,0)$.

What initial value do we take in this case?

Could we pick $(x(0),y(0))=(x_0,1-x_0)$ ? (Thinking)