Multivariate Calculus - Bruce E. Shapiro

66 LECTURE 9. THE PARTIAL DERIVATIVE
Repeating the process, the last equation must equal a constant that we call K 2 so
that
Z ′′ (z) = −K 2 Zz
Y ′′ (y) = (K 2 − K 1 )Y (y) = K 3 Y (y)
whereK 3 = K 2 −K 1 . So each of the three functions X(x), Y (y), Z(z) satisfy second
order differential equations of the form
X ′′ (x) = −kX(x)
which you might recognize as the equation of a ”spring” or ”oscillating string.”
Revised December 6, 2006. Math 250, Fall 2006