PDE Questions! General Solution of Wave-kind Equations I encountered an difficult wave equation plus an extra term which I have no clues how to solve as the following: 
Find positive functions $f$ such that
$\frac{1}{2}\frac{\partial^2 f(x,y)}{\partial x^2}=\frac{1}{2}\frac{\partial^2 f(x,y)}{\partial y^2}+\frac{\partial(y \cdot f(x,y) )}{\partial y}$ on $\mathbb{R}\times[0,y]$.
Since I only want a general solution so the initial conditions and boundary value conditions are omitted.
Cheers
 A: The solution to the equation you mentioned can easily obtained by the method of Separation of Variables.
$$\eqalign{
  & {1 \over 2}{{{\partial ^2}f} \over {\partial {x^2}}} = {1 \over 2}{{{\partial ^2}f} \over {\partial {y^2}}} + {{\partial (yf)} \over {\partial y}} = {1 \over 2}{{{\partial ^2}f} \over {\partial {y^2}}} + \left[ {f + y{{\partial f} \over {\partial y}}} \right]  \cr 
  & f\left( {x,y} \right) = \chi \left( x \right)\Gamma \left( y \right)  \cr 
  & \left[ {{1 \over 2}\Gamma \left( y \right){{{\partial ^2}\chi \left( x \right)} \over {\partial {x^2}}}} \right] = \chi \left( x \right)\left[ {{1 \over 2}{{{\partial ^2}\Gamma \left( y \right)} \over {\partial {y^2}}} + \Gamma \left( y \right) + y{{\partial \Gamma \left( y \right)} \over {\partial y}}} \right]  \cr 
  & {1 \over {\chi \left( x \right)}}\left[ {{1 \over 2}{{{\partial ^2}\chi \left( x \right)} \over {\partial {x^2}}}} \right] = {1 \over {\Gamma \left( y \right)}}\left[ {{1 \over 2}{{{\partial ^2}\Gamma \left( y \right)} \over {\partial {y^2}}} + \Gamma \left( y \right) + y{{\partial \Gamma \left( y \right)} \over {\partial y}}} \right]  \cr 
  & {1 \over {\chi \left( x \right)}}\left[ {{1 \over 2}{{{\partial ^2}\chi \left( x \right)} \over {\partial {x^2}}}} \right] = c  \cr 
  & {1 \over {\Gamma \left( y \right)}}\left[ {{1 \over 2}{{{\partial ^2}\Gamma \left( y \right)} \over {\partial {y^2}}} + \Gamma \left( y \right) + y{{\partial \Gamma \left( y \right)} \over {\partial y}}} \right] = c \cr} $$
As can be seen, the PDE is separated into two Ordinary Differential Equations. First we go through the solution of the first one.
$$\eqalign{
  & {1 \over {\chi \left( x \right)}}\left[ {{1 \over 2}{{{\partial ^2}\chi \left( x \right)} \over {\partial {x^2}}}} \right] = c  \cr 
  & {{{d^2}\chi \left( x \right)} \over {\chi \left( x \right)}} = 2cd{x^2}  \cr 
  & \left( {Ln\chi  + {c_1}} \right)d\chi  = \left( {2cx + {c_2}} \right)dx  \cr 
  & \chi Ln\chi  - \chi  + {c_1}\chi  + {c_3} = c{x^2} + {c_2}x + {c_4}  \cr 
  & \chi \left( {Ln\chi  + {c_5}} \right) = c{x^2} + {c_2}x + {c_6} \cr} $$
As far as we have no information about the boundary conditions, we may simplify our solution by assuming the constants to be zero.
$$\eqalign{
  & \chi Ln\chi  = c{x^2}  \cr 
  & c = {{\chi Ln\chi } \over {{x^2}}} \cr} $$
It is obvious that finding the $\chi$ as a function of $x$ is a little hard or may be not possible analytically and one may need some numerical solutions to do this. And the second equation can also be solved:
$$\eqalign{
  & {1 \over 2}{{{\partial ^2}\Gamma \left( y \right)} \over {\partial {y^2}}} + \Gamma \left( y \right) + y{{\partial \Gamma \left( y \right)} \over {\partial y}} = c\Gamma \left( y \right)  \cr 
  & {{{\partial ^2}\Gamma \left( y \right)} \over {\partial {y^2}}} + 2y{{\partial \Gamma \left( y \right)} \over {\partial y}} + \left( {1 - c} \right)\Gamma \left( y \right) = 0  \cr 
  & \Gamma \left( y \right) = {e^{ - {y^2}}}{c_7}{{\rm{H}}_{ - {1 \over 2}\left( {1 + c} \right)}}\left( y \right) + {e^{ - {y^2}}}{c_8}{}_1{{\rm{F}}_1}\left( {{{1 + c} \over 4};{1 \over 2};{y^2}} \right) \cr} $$
However; one may discuss the values of c and the other boundary conditions.
