Have you read the question correctly? What's the reference? The statement should be that is part of a line through the origin; it doesn't have to be the entire line.
does it necessarily follow from dr/dt = ar(t) that r(t) must trace out a straight line through the origin? r(t) is a vector function and 'a' is a scalar constant.
i have a counterexample: r(t)={2exp(at), 3exp(at)}. r(t) cannot be {0, 0} because of properties of exp(at).
this is a qustion in a calculus book. so i am assuming i am missing something by finding a counterexample.
exact question as written in Richard A. Silverman's Modern Calculus and Analytic Geometry:
suppose the radius vector r (bold vector notation) = r(t) of a moving particle satisfies the differential equation dr/dt (bold vector r) = ar(t) where a is a scalart constant. Prove the particle's trajectory is a straight line through the origin.
I don't see how I've misinterpreted. But Silverman's books are good so I tend to trust.
There seems to be a small ommission in Silverman's question. The solution you found is correct and you're right in that it doesn't pass through the origin. The trajectory is a line through the origin but doesn't include the origin.