Discussion
Let \(u_{1}, u_{2}\) be two solutions of the Cauchy problem :
\(u_{tt} - u_{xx} = x + t^{2}\)
\(u(x, 0) = \cos x,\ u_{t}(x, 0) = 3\)
Then the solutions \(u_{1}\) and \(u_{2}\) satisfy :
Answer: C
The two solutions satisfy \(u_{1} - u_{2} \equiv 0\).
The difference \(w = u_{1} - u_{2}\) solves the homogeneous problem \(w_{tt} - w_{xx} = 0\) with \(w(x, 0) = 0\) and \(w_{t}(x, 0) = 0\). By d'Alembert's formula (or the energy method), the only solution is \(w \equiv 0\). So the Cauchy problem for the wave equation has a unique solution and \(u_{1} = u_{2}\).
With \(u_{1} = u_{2} = u\), \(u_{1} = 2u_{2}\) and \(u_{1} + u_{2} = 0\) would force \(u \equiv 0\), which does not meet \(u(x, 0) = \cos x\), and \(u_{1}u_{2} = u^{2} = 1\) fails too.
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