For xu_x+yu_y=u with data u(t,t)=f(t),t>0, consider: P:if f(t)=2t+1 there is a unique local solution near the initial curve. Q:if f(t)=2t−1 there are infinitely many local solutions. Choose truth values.
Differentiate the data along the curve: f′(t)=u_x(t,t)+u_y(t,t). The PDE there requires tf′(t)=f(t). Both proposed functions violate this, since 2t≠2t±1. Neither has a differentiable solution near the curve. **D**.
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