Discussion
Consider the Cauchy problem
\[ \begin{aligned} x\frac{\partial u}{\partial x} &+ y\frac{\partial u}{\partial y} \\ &= u; \end{aligned} \]
u = f(t) on the initial curve \(\Gamma = (t,t)\); \(t \gt 0\).
Consider the following statements:
P: If f(t) = 2t + 1, then there exists a unique solution to the Cauchy problem in a neighbourhood of \(\Gamma\).
Q: If f(t) = 2t - 1, then there exist infinitely many solutions to the Cauchy problem in a neighbourhood of \(\Gamma\).
Then
Answer: D
Both P and Q are FALSE.
The characteristic equations are \(\frac{dx}{x} = \frac{dy}{y} = \frac{du}{u}\). The characteristic curves are rays through the origin, y/x constant, along which u grows linearly with the distance: every solution is homogeneous of degree 1, \(u = x\,g(y/x)\).
The initial curve \(\Gamma\), the line y = x, is itself a characteristic, so the Cauchy problem is characteristic. Along it, any solution must satisfy u(t, t) = t g(1) = ct for a constant c. A solution exists only if the data f(t) has this form, and then there are infinitely many (g is free away from the ray).
Neither 2t + 1 nor 2t - 1 is a constant multiple of t, so in both cases there is no solution at all: no unique solution for P and no infinitely many for Q.
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