Consider the Cauchy problem
\(u \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=1\), (x, y) ∈ ℝ × (0, ∞),
u(x, 0) = kx, x ∈ ℝ
with a given real parameter k. For which of the following values of k does the above problem have a solution defined on R × (0, ∞)?
1
k = 0
2
k = -2
3
k = 4
4
k = 1