**Unformatted text preview: **a and f are periodic with period P : u 00 ( x + P ) + a ( x ) u ( x + P ) = f ( x ) for all x . Thus, if we let v ( x ) : = u ( x + P ) , then v ( x ) + a ( x ) v ( x ) = f ( x ) . But if u ( P ) = u ( ) , then v ( ) = u ( ) . Therefore by the uniqueness assertion, v ( x ) = u ( x ) for all x , that is, u ( x + P ) = u ( x ) for all x . The related assertion for a solution of a second order equation is essentially identical except there we need to assume that both u ( ) = u ( P ) and u ( ) = u ( P ) , since the corresponding uniqueness assertion for second order equations requires that. 1...

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