Solving an Inhomogeneous $1$st Order PDE using Method of Characteristics

Question

I wanna solve the equation

$$u_x+u_y+u=\exp(x+2y), \quad u(x,0) = 0$$

I have just learned method of characteristics. But I don't know how to deal with $u$ term and inhomogeneous term simultaneously. Can you help me?

Answer 1

Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dy}{dt}=1$ , letting $y(0)=0$ , we have $y=t$

$\dfrac{dx}{dt}=1$ , letting $x(0)=x_0$ , we have $x=t+x_0=y+x_0$

$\dfrac{du}{dt}=e^{x+2y}-u=e^{3t+x_0}-u$ , we have $u(x,y)=\dfrac{e^{3t+x_0}}{4}+f(x_0)e^{-t}=\dfrac{e^{x+2y}}{4}+f(x-y)e^{-y}$

$u(x,0)=0$ :

$f(x)=-\dfrac{e^x}{4}$

$\therefore u(x,y)=\dfrac{e^{x+2y}-e^{x-2y}}{4}$