TITLE: Mysterious step while solving ODE
QUESTION [4 upvotes]: I just read a paper in which an ODE was solved using a step I don't understand.  
$${dI \over \kappa (x) dx} = I$$ 
Let
$$\tau = \int_0^x \kappa(x)dx$$
Rewrite equation as
$${dI \over d\tau} = I$$ 
and solve.  
How does the final expression follow from the definition of $\tau$?

REPLY [4 votes]: Chain rule:
$$\frac{dI}{dx}=\frac{dI}{d\tau}\frac{d\tau}{dx}=\frac{dI}{d\tau}\kappa(x).$$

REPLY [4 votes]: If $\tau(x) = \displaystyle \int_{0}^x \kappa(y) dy$, then $d \tau = \kappa(x) dx$
{"set_name": "stack_exchange", "score": 4, "question_id": 24949}