1D Heat Equation with Inhomogeneous Neumann Boundary Conditions

Question

I'm trying to solve the 1D heat equation with inhomogeneous boundary conditions.

$u_t=Ku_{xx}$

$u_x(0,t)=F_0 \hspace{0.2cm} and\hspace{0.2cm} u_x(L,t)=F_1$

$u(x,0)=f(x)$

My first step was to write the solution in two parts:

1. $U(x)$, which satisfies the time-independent heat equation
2. $\hat{u}(x,t)$, which will produce the correct homogenous Neumann conditions

So,

$u(x,t) = U(x) + \hat{u}(x,t)$

Since $U(x)$ has no dependence on time, $U_t=0$, so impose that

$0=KU_{xx}$

$U_x(0)=F_0 \hspace{0.2cm} and\hspace{0.2cm} U_x(L)=F_1$

This is where I get stuck when I solve for the steady-state solution I get $F_0=F_1$. Looking around on the internet I have come across using a particular solution instead of a steady-state solution where

$U(x,t) = \frac{F_0-F_1}{2L}x^2-F_0x+\frac{K^2(F_0-F_1)}{L}t$

However, I do not understand this step, or how they find that particular solution. I was hoping someone could explain?