Calculating the value of $\frac {1}{D^2+a^2} \sin ax$ where the operator $D = \frac {d}{dx}$. Error in Method $1$?

Question

Calculating the value of $\dfrac {1}{D^2+a^2} \sin ax$ where the operator $D = \dfrac {d}{dx}$

I have tried calculating using two different methods and both of them yield different results. Could someone please pinpoint any error in Method $1$ which I might be making. My textbook uses method $2$. Thanks a lot for your help

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Answer 1

These solutions are equivalent. Their difference is a solution of the homogeneous equation, so both are suitable particular solutions. None of them are special in the sense that both value and first derivative at zero are zero.