The author solves the well-known Mellin convolution equation \[ \int_0^\infty u\left(\frac xy\right)f(y)\frac{dy}y=g(x) , \qquad x>0\tag{1} \] where \(g\) is a prescribed function, \(f\) is unknown and the kernel \(u\) is a particular function \[ u(x)= [x^l(k+x\partial_x)]^n \{(ax^p+b)^{\alpha +\gamma n}(cx^q+d)^{\beta+\delta n}e^{-px^\eta}\}. \] Actually, equation (1) is solved simply: the Mellin transform \[ H(s)=M\big[h(x);s\big]:=\int_0^\infty x^sh(x)\frac{dx}x, \] transforms the equation into the trivial equality \(U(s)F(s)=G(s)\), which is solved by the inverse Mellin transform. The author finds more or less explicit expression for the kernel transform \(U(s)=M\big[u(x);s\big]\) and considers several particular cases of the kernel, treated earlier by other authors.