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.



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.


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