The PDE $$\frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2}-V_0\frac{\partial u}{\partial x}$$
can be separated into two ODEs by the method of separation of variables, and the ODE for $M(x)$ is $$\frac{d^2M}{dx^2}(x)-\frac{V_0}{D}\frac{dM}{dx}(x)=-\frac{\lambda}{D} M(x)$$
If we now multiply this equation by the integrating factor $$\exp\left(-\int \frac{V_0}{D}dx\right)$$ we'll get the equation $$\left[M'(x)\exp\left(-\int \frac{V_0}{D}dx\right)\right]'=-\frac{\lambda}{D}\exp\left(-\int \frac{V_0}{D}dx\right)M(x)$$
Why is this not an equation of the Sturm-Lioville form?