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I am asked to find the adjoint of the following differential operators:

\begin{equation} L_{1} = a(x)\frac{\partial}{\partial x}b(x)\frac{\partial}{\partial x}, \end{equation}

and

\begin{equation} L_{2} = \frac{\partial}{\partial t} + b(x)\frac{\partial}{\partial x}, \end{equation}

From my understanding

\begin{equation} adj(\frac{\partial}{\partial t}) = -\frac{\partial}{\partial t} \end{equation}

and

\begin{equation} adj(\frac{\partial^2}{\partial t^2}) = \frac{\partial^2}{\partial t^2} \end{equation}

Can somebody please explain how to find the adjoint of the two operators? Thank you!

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