Prompt Detail:
You are correct! I apologize for the mistake in my previous response. The correct integration of dx/(-x+t) is:
∫dx/(-x+t) = - ln|-x+t| + C
Substituting this into our differential equation, we have:
-ln|-x+t| = ln|y-t| + C
Simplifying, we get:
|-x+t| = e^C|y-t|
Squaring both sides, we get:
(x-t)^2 = e^(2C)(y-t)^2
This is the equation for the streamlines of the flow. It represents a family of ellipses centered at (t, t) with major axis along the x-axis and minor axis along the y-axis. The value of the constant e^C determines the shape and size of the ellipse, with larger values of e^C corresponding to larger ellipses.
Again, I apologize for the error in my previous response. Thank you for bringing it to my attention.
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