I am going through a book on partial differential equations, and am slightly confused by this paragraph:
Consider the equation $\frac{dy}{dx}=p(x, y)$.
a) Think of solutions to this equations as curves in the $xy$ plane. The equation is telling us that the slop of the tangent line to a solution curve at the point $(x_0, y_0)$ is equal to $p(x_0, y_0)$.
b) We can also say that the solution curve at the point $(x_0, y_0)$ is changing in the direction of the vector $(1, p(x_0, y_0))$.
Where does the vector $(1, p(x_0, y_0))$ come from? Is this simply the gradient?
If $m$ is the slope of a tangent line to a curve, then the tangent vector of the curve is parallel to the $(1,m)$ vector in the $(x,y)$-plane. Think of it as going back to the "rise over run" ($\frac{\Delta y}{\Delta x}$) idea: We move $1$ unit in $x$ and $m$ units in $y$. Here $m= p(x_0,y_0)$.