Solve the following nonhomogeneous heat conduction problem using eigenfunction expansion.

1. \(\displaystyle{ u_t=u_{xx}-x, \quad u(0,t)=0, \quad u_x(1,t)=0, \quad u(x,0)=\sin(\pi x/2)}\)  solution

2. \(\displaystyle{ u_t=u_{xx}+xe^{-t}, \quad u(0,t)=0, \quad u(1,t)=0, \quad u(x,0)= 5}\)  solution

3. \(\displaystyle{ u_t=u_{xx}+e^{-t}(1-x), \quad u(0,t)=0, \quad u_x(1,t)=0, \quad u(x,0)=0}\)    solution

4. \(\displaystyle{ u_t=u_{xx}+xe^{-t}, \quad u(0,t)=0, \quad u_x(1,t)+u(1,t)=0, \quad u(x,0)= 0}\)  solution