solution of multidimensional PDE I'm looking for a way to find a solution 'f' to the following PDE.
$$ y \frac{\partial f}{\partial r} + g_1(r)\left(z\frac{\partial f}{\partial y} - y\frac{\partial f}{\partial z}\right) + g_2(r)\left(y\frac{\partial f}{\partial x} - x\frac{\partial f}{\partial y}\right) = 0$$
$r\in [a,b]$, x, y and z are $]-\infty, \infty[$
I know $f(a,x,y,z)$ and $f(b,x,y,z)$. I also know that $f\to\infty$ for $x,y,z \to \pm \infty$, so I can impose $f(x_{\max})=f(y_{y\max}) = f(z_{z\max}) = 0$
The functions $g_1$ and $g_2$ are  
$g_1(r) = \tanh(r)$
$g_2(r) = 1/\cosh(r)$
 A: $y\dfrac{\partial f}{\partial r}+\tanh(r)\left(z\dfrac{\partial f}{\partial y}-y\dfrac{\partial f}{\partial z}\right)+\text{sech}(r)\left(y\dfrac{\partial f}{\partial x}-x\dfrac{\partial f}{\partial y}\right)=0$
$y\dfrac{\partial f}{\partial r}+y~\text{sech}(r)\dfrac{\partial f}{\partial x}+(z\tanh(r)-x~\text{sech}(r))\dfrac{\partial f}{\partial y}-y\tanh(r)\dfrac{\partial f}{\partial z}=0$
$\dfrac{\partial f}{\partial r}+\text{sech}(r)\dfrac{\partial f}{\partial x}+\dfrac{z\tanh(r)-x~\text{sech}(r)}{y}\dfrac{\partial f}{\partial y}-\tanh(r)\dfrac{\partial f}{\partial z}=0$
Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dr}{dt}=1$ , letting $r(0)=0$ , we have $r=t$
$\dfrac{dx}{dt}=\text{sech}(r)=\text{sech}(t)$ , letting $x(0)=x_0$ , we have $x=x_0+\tan^{-1}\sinh(t)=x_0+\tan^{-1}\sinh(r)$
$\dfrac{dz}{dt}=-\tanh(r)=-\tanh(t)$ , letting $z(0)=z_0$ , we have $z=z_0-\ln\cosh(t)=z_0-\ln\cosh(r)$
$\dfrac{dy}{dt}=\dfrac{z\tanh(r)-x~\text{sech}(r)}{y}=\dfrac{z_0\tanh(t)-x_0\text{sech}(t)-\tanh(t)\ln\cosh(t)-\text{sech}(t)\tan^{-1}\sinh(t)}{y}$ , letting $y(0)=y_0$ , we have $\dfrac{y^2}{2}=\dfrac{y_0^2}{2}+z_0\ln\cosh(t)-x_0\tan^{-1}\sinh(t)-\dfrac{(\ln\cosh(t))^2}{2}-\dfrac{(\tan^{-1}\sinh(t))^2}{2}=\dfrac{y_0^2}{2}+z\ln\cosh(r)-x\tan^{-1}\sinh(r)+\dfrac{(\ln\cosh(r))^2}{2}+\dfrac{(\tan^{-1}\sinh(r))^2}{2}$
$\dfrac{df}{dt}=0$ , letting $f(0)=F(x_0,y_0^2,z_0)$ , we have $f(r,x,y,z)=F(x_0,y_0^2,z_0)=F(x-\tan^{-1}\sinh(r),2x\tan^{-1}\sinh(r)+y^2-2z\ln\cosh(r)-(\tan^{-1}\sinh(r))^2-(\ln\cosh(r))^2,z+\ln\cosh(r))$
