The equation $xu_{xx}(x,y)+2u_x(x,y)+Au_{yy}(x,y)=f(x,y)$, $f(x,y)=\sum^\infty_{n=0}a_n(y)l_n\in E(\mathbf{R_+},LG'_0)$ ( resp. $E(\mathbf{R_+},LG'_e)$), $A\in\mathbf R$ is considered. We solve the corresponding boundary value problem in $E(\mathbf{R_+},LG'_e)$ (Prop. 1). If $f$ has the expansion in $x$ of appropriate form and $A$ is a smooth function or a constant than we find the Laguerre series solution (Prop. 2). If $A$ is a smooth function on $(0,\infty)$ we solve this equation in $E(\mathbf{R_+},LG'_0)$, (resp. $E(\mathbf{R_+},LG'_e)$), assuming that an, $a_n\in\mathbf{N_0}$, and the first coefficient of the Laguerre series solution is the polynomial of arbitrary but fixed degree (resp. of first degree).