# General solution for ''homogeneous and linear PDE''

Consider function $$u:[0,T]\times \mathbb{R}\rightarrow\mathbb{R}$$ that satisfies the following partial differential equation: $$u_t(t,x)=au(t,x)-bu_x(t,x)-cu_{xx}(t,x) \quad \text{with} \quad u(T,x)=1,$$ where $$a,b,c\in\mathbb{R}$$.

My attempts: By Separation of variables (https://en.wikipedia.org/wiki/Separation_of_variables), let $$u(t,x)=S(x)T(t)$$ where $$S(x)$$ and $$T(t)$$ are functions. Then the PDE becomes: $$\frac{T'(t)}{T(t)}=a-b\frac{S'(x)}{S(x)}-c\frac{S''(x)}{S(x)}.$$ Since both sides depends on seperate variables, then they must equaly to some constants say $$-\lambda$$. Then we obtain the ODEs: $$\frac{T'(t)}{T(t)}=-\lambda$$ and $$a-b\frac{S'(x)}{S(x)}-c\frac{S''(x)}{S(x)}=-\lambda.$$ Suppose $$\lambda>0$$, then $$T(t)=e^{-\lambda (T-t)}$$. Then the terminal condition implies $$S(x)T(T)=S(x)=1$$, which means $$S(x)=1$$ for all $$x\in\mathbb{R}$$ and $$u(t,x)=e^{-\lambda (T-t)}$$. However, this is a trivial solution. I wonder what approach would yield a general solution to this specific PDE?