How can I animate the plot of solutions to this system with different initial conditions?

Question

I would like to plot the curves given by $\alpha(s) = (u(s), v(s))$, where $u$ and $v$ are like below.

Clear[u, v];
{u, v} = {u[t], v[t]} /. 
   NDSolve[{u'[t]^2 + v'[t]^2 == 1, 
      u'[t] v''[t] - u''[t] v'[t] == u[t] v'[t] - v[t] u'[t], 
      u'[0] == Sin[0.18], v'[0] == Cos[0.18]}, {u[t], v[t]}, {t, -7.5,
       7.5}][[1]];

What I want to do is, for each different initial conditions $u'(0) = \cos(\theta)$, $v'(0) = \sin(\theta)$, $u'(t)v''(t) - u''(t)v'(t) = -c(u'(t)v(t) - u(t)v'(t))$ with $\theta$ varying in $[0, 2\pi]$ and $c$ varying in, say, $(0, 10]$, plot a new curve. I'm new to Mathematica so I still have a hard time doing these little things.

Answer 1

ClearAll[u, v, θ, c]
pndsv = ParametricNDSolveValue[{u'[t]^2 + v'[t]^2 == 1, 
    u'[t] v''[t] - u''[t] v'[t] == -c (-u[t] v'[t] + v[t] u'[t]), 
    u'[0] == Cos[θ], v'[0] == Sin[θ]}, {u, v}, {t, -7.5, 7.5}, {c, θ}] 

An alternative visualization of the solution:

Manipulate[ParametricPlot[Evaluate@Through@pndsv[c, θ][t], {t, -7.5, 7.5}, 
    PlotRange -> {{-3, 3}, {-3, 3}}], {c, 0, 10}, {θ, 0, 2 Pi}]

Answer 2

Use ParametricNDSolveValue instead:

sol=ParametricNDSolveValue[
    {
    u'[t]^2 + v'[t]^2 == 1,
    u'[t] v''[t] - u''[t] v'[t] == c (u[t] v'[t] - v[t] u'[t]),
    u'[0] == Sin[θ],v'[0] == Cos[θ]
    },
    {u,v},
    {t, -7.5, 7.5},
    {c, θ}
];

Then, you can use Manipulate to explore the plot for different values of the parameters:

Manipulate[
    Plot[{sol[c,θ][[1]][x],sol[c,θ][[2]][x]}, {x,-7.5,7.5}],
    {{c, 1}, 0, 10},
    {{θ, .18}, 0, 2Pi}
]