non-separable "heat equation"

Question

I am trying to solve the following PDE: $$ \frac{\partial f(t,x)}{\partial t}=\frac{\partial^2 f(t,x)}{\partial x^2}x(1-x) $$ with $(t,x)\in [0,1]^2$ and boundary conditions $f(1,\cdot)=f(\cdot,1)=0$. Is there a non-trivial solution to this? I tried to separate the variables but one of the boundary conditions gets violated in that case. Specifically, if we guess a non-trivial solution takes the form $f(t,x)=F(t)G(x)$, then the PDE becomes $$ \frac{F'(t)}{F(t)}=\frac{G''(x)}{G(x)}x(1-x)=-c $$ which must hold for any $(t,x)$ pair in the interior of domain. Hence, the fractions equal some real number $-c$. The general solutions for the resulting ODEs are $$F(t)=Ae^{-ct}$$ and $$G(t)=Bx^y(1-x)^z$$ where $y,z$ are constants that depend on $c$ and $A, B, c$ are yet to be determined constants. But then $$f(1,x)=ABe^{-c}x^y(1-x)^z \neq 0$$ for $x<1$ unless $AB=0$ which makes the solution trivial.