Combining two differential to make into one.

Question

By using equation 6 and 7 the author formed another differential equation.But I did not understand how he did.

Answer 1

You have a structure $$ p_1(D)U+q(D)W=0\\ p_2(D)W-q(D)U=0 $$ with $D$ the derivation operator, $DU=U'$, $D^2U=U''$ etc. and $p(D)$, $q(D)$ the linear operators that are polynomials in $D$. \begin{align} p_1(D)&=EI_{zz}D^4-2k^2(2EI_{xx}+EI_{zz})D^2+(k^4EI_{zz}-m\omega^2)I\\ p_2(D)&=EI_{xx}D^4-2k^2(EI_{xx}+2EI_{zz})D^2+(k^4EI_{xx}-m\omega^2)I\\ q(D)&=2k(EI_{xx}+EI_{zz})(D^3-k^2D) \end{align}

Obviously then one can combine $$ p_2(D)p_1(D)U=-p_2(D)q(D)W=-q(D)p_2(D)W=-q(D)^2U $$ so that $[p_1(D)p_2(D)+q(D)^2]U=0$ and similarly $[p_1(D)p_2(D)+q(D)^2]W=0$. As $p_k$ are both even polynomials and $q$ is odd, the above combination is again an even polynomial, that is, only even order derivatives occur in the resulting equation.