Finding streamlines from velocity components

[CMJ96]

Homework Statement

Homework Equations

$$\frac{dy}{dx}=\frac{v_y}{v_x}$$[/B]

The Attempt at a Solution

I have subbed the given values for $$v_y$$ $$v_x$$ into the equation above, and integrated, i got the following expression
$$y=\left(1+t \right) lnx $$
I'm not sure where to go next, do I sub in $$x_0$$ $$y_0$$ ?

 

[Orodruin]

You cannot solve it like that. Your differential equation depends on time $t$, which is not constant. You need to solve the system
$$
\frac{dx}{dt}= v_x, \quad \frac{dy}{dt}= v_y.
$$

Edit: Sorry, missed the title. Those are the pathlines. For the streamlines yes, but you are missing an integration constant that you will need to adapt your solution to the initial condition. Otherwise you would have only one streamline.

 

[CMJ96]

So I should sub in limits of x=x_0 and y=y_0 up to y=y and x=x and add in a constant of integration?