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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1993 • Question 10

Question

Math 6410 • ODE Fall 1993 • Question 10

Let \(U:\mathbb {R}^2\longrightarrow \mathbb {R}\) be given by \[ U(x,t)=f(x+t)+g(x-t), \] where \(f,g\in L^1_{\operatorname {loc}}(\mathbb {R})\). Show that \[ \partial _x^2 U=\partial _t^2 U \] in the sense of distributions.

Raw LaTeX

Let \(U:\R^2\longrightarrow\R\) be given by
\[
U(x,t)=f(x+t)+g(x-t),
\]
where \(f,g\in L^1_{\operatorname{loc}}(\R)\). Show that
\[
\partial_x^2 U=\partial_t^2 U
\]
in the sense of distributions.

Topic Tags

Distributions and Weak Solutions