# 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

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.