$(10$ points$)$

$(a)(5$ points$)$

$\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}+3x\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}=1+u,-0$

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

$(b)(5$ points$)$

$t\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}+2x\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}=1,-1$

$u(x,1)=f(x)$