**Unformatted text preview: **u (0 ,t ) = 0 u x (3 ,t ) = 0 for all t > 0. (Hint: Look for “separated” solutions.) 4. Find the closed form (similar to d’Alembert’s formula) of the solution u ( x,t ) of the initial-boundary value problem for the semi-inﬁnite string: u tt-c 2 u xx = 0 for x,t > where u ( x, 0) = f ( x ) for x > 0, and u t ( x, 0) = 0 for x > 0, and u (0 ,t ) = α ( t ) for t ≥ 0, where f and α are C 2 functions and satisfy f (0) = α (0), α (0) = 0 and α 00 (0) = c 2 f 00 (0). Verify that the solution is C 2 for all x,t > 0....

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