Question

Reduce the equation to a set of ODDE. using separation of variable

(del)²A+[k²+f(p)+1/p² g(B)+h(z)]A =0

Answer 1

$\displaystyle (del)²A+[k²+f(p)+1/p² g(B)+h(z)]A =0\\ \textsf{The auxiliary equation is}\\ m^2 = -\left(k²+f(p)+1/p² g(B)+h(z)\right)\\ m = \pm i\sqrt{k²+f(p)+1/p² g(B)+h(z)}\\ \therefore A = C_1\cos\left(\sqrt{k²+f(p)+1/p² g(B)+h(z)}\right) + C_2\cos\left(\sqrt{k²+f(p)+1/p² g(B)+h(z)}\right)$