text a u_tc u_x0 ux 0x x in mathbbr t geq 0 text b u_t t u_xe t ux 0si

Question

3. Solve the following first-order PDE using the method of characteristics, and plot the characteristics on the x-t plane: \text { (a) } u_{t}+c u_{x}=0, u(x, 0)=x, x \in \mathbb{R},

t \geq 0 \text { (b) } u_{t}-t u_{x}=e^{-t}, u(x, 0)=\sin (x), x \in \mathbb{R}, t \geq 0 \text { (c) } u_{i}-3 u u_{x}=0, u(x, 0)=x, x \in \mathbb{R}, t \geq 0