So I tried to solve this PDE by separation of variables which was not applicable because of non-homogeneous nature of this PDE. Then I assumed ( after reading from a book ) :
$$ T(x,y)=v(x,y) + \phi(x) $$
After this, I solved the equation and got
$$ v(x,y)=0$$ and $$ \phi(x)=-x^2+5.5x $$
Please tell me the error and give me the direction to solve this problem myself.