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DFA and Reg Exp

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$ L = \{ wxwy \ | \  \  x,y,w \ \  \epsilon\  ( a + b )^+ \} $

Draw the DFA and also write the Reg exp for the above language.
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Tuhin Dutta asked Dec 4, 2017
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Simple doubt in Reg Exp of TOC
why $bb^*$ is $b^*$ and not $b^+$? Ref: $a^*(bb^*a + a)a^*\\=a^*(bb^*+\epsilon)aa^*\\=a^*b^*aa^*\\=a^*b^*a^*a$
why $bb^*$ is $b^*$ and not $b^+$?Ref:$a^*(bb^*a + a)a^*\\=a^*(bb^*+\epsilon)aa^*\\=a^*b^*aa^*\\=a^*b^*a^*a$
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