I was reading about solving elliptic equations. The example given is:
$$yu_{xx}+u_{yy}=0$$
where $y>0$.
Via the characteristic equation, the example comes to the substitutions:
$$\zeta=2y^{3/2}; \eta=3x$$
The example then says that by substitution we arrive at:
$$u_{\zeta\zeta}+u_{\eta\eta}=-\frac{u_\eta}{3\zeta}$$
But I don't see how to make this last step. I see that:
$$u_{xx}=9u_{\eta\eta}$$
However, how do we work out $u_{yy}$ in term of $\zeta$?
Probably there is a mistake and
$$\zeta=2y^{3/2}; \eta=3x$$
Using the chain rule:
$$u_y=u_{\zeta}\zeta_y+u_{\eta}\eta_y=u_{\zeta} 3\sqrt y$$
Using the chain rule again.. (Notice that we have to use the product rule)
$$ u_{yy}= \frac{3}{2\sqrt y}u_{\zeta} + 3\sqrt y (u_{\zeta \zeta}3\sqrt y)=\frac{3}{2\sqrt y}u_{\zeta}+ 9y u_{\zeta \zeta} $$
Then:
$$yu_{xx}+u_{yy}= 9yu_{\eta \eta} + \frac{3}{2\sqrt y}u_{\zeta}+ 9y u_{\zeta \zeta} = 0 $$
So
$$u_{\eta \eta} + u_{\zeta \zeta}=\frac{-3u_{\zeta}}{18y^{\frac{3}{2}}}$$
Notice that $y^{\frac{3}{2}}= \frac{\zeta}{2}$