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I have this following PDE to solve for $u(x,t): x\in(-\infty,\infty), t\in(0,T)$

$$u_t+\frac{1}{2}u_{xx} = f(x)u$$ Where $f(x) = \gamma x^2$.

I purposed the Boundary conditions: $$u(\infty, t) = u(-\infty, t) = 0, u(x, 0) = C, u(x,T) = 1$$

Would this be solvable? I just have no idea how to work with $\infty$.

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