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**I'm using the method of characteristics to solve a pde of the from ## au_{x}+bu_{y}=c##**

*where ## a=\frac{dx}{d \tau} , b= a=\frac{dy}{d \tau}, c=a=\frac{du}{d \tau}##*

where initial data is parameterised by ##s## and initial curve given by ##x( \tau)=x_{0}(s)##, ##y( \tau)=y_{0}(s)## and ##u( \tau)=u_{0}(s)## at ## \tau = 0## .

where initial data is parameterised by ##s## and initial curve given by ##x( \tau)=x_{0}(s)##, ##y( \tau)=y_{0}(s)## and ##u( \tau)=u_{0}(s)## at ## \tau = 0## .

**Question:**

Questions been given so far on my course have ## c ## given explicitly. I'm trying to solve:

##u_{x} + u_{y} =f(x,y)##. subject to initial data ##u=g(y)## on ##x=0##, functions ##f## and ##g## given.

**Attempt:**

Solving ## \frac{du}{d\tau} = f(x,y)##

i get ##u= \int f(x(\tau),y(\tau))d\tau+ u_{0}.##

**QUESTION:**In the solution the limits of this integral run from ##\tau## to 0. I'm not used to doing these with limits, and if I'm honest, I'm totally clueless where this comes from... can someone please explain the integral limits to me? usually when it's just given explicitly I just integrate.

So the solution has : ##u= \int^{\tau} _{0} f(x(\tau '),y(\tau ')) d\tau '+ u_{0}##

**Many thanks in advance.**

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