the problem
$$
\begin{cases}
u_t(x,t)=u_{xx}(x,t)+2\\
u_x(0,t)=1\\
u_x(1,t)=-1\\
u(x,0)=f(x)
\end{cases}
0<x<1, 0<t
$$
my approach
$$
u(x,t)=A(x,t)+B(x,t)\\
A:homogeneous, B:non-homogeneous\\
\text{then }\\
B_x(0,t)=1\\
B_x(1,t)=-1\\
\text{and let }B(x,t)=x-x^2\\
\text{then }\\
\begin{cases}
A_t=A_{xx}\\
A_x(0,t)=0\\
A_x(1,t)=0\\
A(x,0)=f(x)-x+x^2
\end{cases}
$$
is it right approach?
if it is, could you give me exact solution?
thanks for your help