I want to compute the plane curves with curvature:
$\kappa (s) = 5/(s+1)$
I know I must apply the fundamental theorem of curves and solve some differential equation, but I am a bit confused about how to apply that theorem. Could you help with what equation must I solve and why?
Consider the polar angle $\theta$ of a tangent vector. You can see that $\theta'(s) = \kappa(s)$. Thus you can solve ODE:
$$ \theta'(s) = \frac 5{s+1},\qquad\theta(0)=\theta_0,\\ \theta(s) = \theta_0 + 5\ln (s+1). $$
Then you can use the fact that tangent vector $(dx,\ dy) = (ds\cos\theta,\ ds\sin\theta)$. In other words, you have a system of independent equations:
$$ x'(s) = \cos\Big(\theta_0+5\ln(s+1)\Big),\qquad x(0)=x_0,\\ y'(s) = \sin\Big(\theta_0+5\ln(s+1)\Big),\qquad y(0)=y_0, $$
For brevity, I will use $\theta_0=0$, but it doesn't make the integral any harder:
$$ x(s) = \frac{s+1}{26}\Big(\cos \theta(s) + 5 \sin \theta(s) \Big)-\frac{1}{26} + x_0,\\ y(s) = \frac{s+1}{26}\Big(\sin \theta(s) - 5 \cos \theta(s) \Big)+\frac{5}{26} + y_0 $$
You have 3 degrees of freedom here: 2 for the location of a starting point $x_0,y_0$ and 1 for the arbitrary rotation of the curve $\theta_0$.