I know the heat equation is $$\frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)$$
I know that $u(t,x)$ is the temperature distribution at time $t$ at the point $x$.
We assume that the temperature at $x=0$ and $x=1$ is fixed.
In the notes then it says
"consider $u(t,x) \to u(t,x)-a-(b-a)x$ and it is clear that $a=b=0$"
Why do we consider this and where does $u(t,x)-a-(b-a)$ come from?
And how can we see $a=b=0$?
Because of the form of the heat equation you can always add an arbitrary first-degree polynomial in $x.$ That freedom is used to satisfy the inhomogeneous boundary conditions (temperature at the end points known but not necessarily zero) when a hypothetical solution with homogeneous boundary conditions $t(0)=t(1)=0$ is supposed known.