The solution looks different from the cartoon…
Anyway, you can plot the solution as two parts, one of which has an arrow:
x1(t) = exp(-2t)*cos(t)
x2(t) = exp(-2t)*sin(t)
plot(x1, x2, 0, π/6, legend=false, color=:black, arrow=true)
plot!(x1, x2, π/6, 2π, legend=false, color=:black)