I was thinking about the problem that says:
The initial value problem $u_{x}+u_{y}=1,u(s,s)=1,0\leq s\leq1,$ has
(a) two solutions,
(b) a unique solution,
(c) no solution,
(d) infinitely many solutions.
My attempts: By using Lagrange's method, we see $\frac{dx}{1}=\frac{dy}{1}=\frac{du}{1}.$ Hence we get, $x=y+c_1$ and $u=x+c_2.$ Now since $u(x,y)=x+c_2 ,$ we get by the given condition that $u(s,s)=1=s+c_2$ and so $c_2=1-s.$ So, finally we get, $u(x,y)=x+(1-s).$ Since $0\leq s\leq1,$we get infinitely many solutions for various choice of s. Am i going in the right direction? Please help.Thanks in advance for your time.