# symmetry of partial differential equations (Heat equation)

Morning everyone, I am doing some problem sheets for my class in Partial differential equations where we dont have an actual textbook. we are given a pack on notes. I am having an issue discerning what is meant by symmetry in the PDE's. We have first that: One of these eigenvalue problems is symmetric and the other isn't.

\begin{align} X'' - aX &= -\lambda X \quad \text{in } (0,\ell) \\ \text{with } X(0) &= X(\ell) = 0 \tag{1} \end{align}

\begin{align} X'' - aX' &= -\lambda X \quad \text{in } (0,\ell) \\ \text{with } X(0) &= X(\ell) = 0 \tag{2} \end{align}

My problem comes next, in the problems we are asked to show that the equation in $$(1)$$ is symmetric and that the one in $$(2)$$ is not. Firstly, I would like to know what symmetry is in the case of these types of equations and then how to show it. Thank you

• I believe symmetry in this case means $X(x) = X(l-x)$. Then the second problem isn't symmetric due to the $X'$ term – Dylan Mar 18 at 13:14