Finding field lines given a plane vector field

Question

Problem: Determine the field lines of the vector field \begin{align*} \mathbf{F}(x,y) = x\hat{i} + y \hat{j}. \end{align*}

The field lines satisfy the system \begin{align*} \frac{dx}{x} = \frac{dy}{y}. \end{align*} Integrating gives \begin{align*} \ln|x| + A = \ln|y| + B, \end{align*} where $A$ and $B$ are constants.

I'm not sure how to proceed now, any help?

Edit: Exponentiating both sides, we get \begin{align*}e^{\ln(x)} e^A = e^{\ln(y)} e^B, \end{align*}, and so \begin{align*} x e^{A-B} = y. \end{align*} Can I now replace $e^{A-B}$ by another constant $C$, and conclude that the field lines are straight lines?

Answer 1

Exponentiate on both sides and use $e^{A+ln|y|}=e^Ay$. Absolute value can be omitted because an exponential of any real number is positive.