I'm trying to understand why the line of slope y passing through (x,y) is an integral manifold. My intuition tells me that there exists a point in the slope field where the distribution cannot be defined, which makes me think that it can not be an integral manifold. However, I'm told that this isn't the case! Could someone please explain how to approach this problem/proof and why I'm wrong?
Thank you very much for your time!
As I understand the question, you are given the non-singular distribution $D$ on $\Bbb R^2$ defined by $$D(x,y)=\mathrm{Vect}\left(\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}\right)\subset T_{(x,y)}\Bbb R^2$$ (it is non-singular because it has constant rank, i.e. dimension, equal to 1) and are asked whether it is integrable. A line field is always integrable, this follows from the existence theorem for integrals to vector fields. For instance you could look at the vector field $X$ defined at every point by $$X(x,y)=\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}$$ An integral curve will be a curve $c:I\to\Bbb R^2$ such that for all $t\in I$, $\dot{c}(t)=X(c(t))$ i.e. $$\begin{cases}\dot{c}_x(t)=1\\\dot{c}_y(t)=c_y(t)\end{cases}$$ which means that the integral curve passing through $(x_0,y_0)$ has $$\begin{cases}c_x(t)=x_0\\c_y(t)=y_0e^t\end{cases}$$ This gives you a parametrization of all the integral manifolds of $D$.