Dtyjlui

Does anybody know how to solve the following differential equation?

$$w^2cdot r+frac{dw}{dt}cdot r+frac{dr}{dt}cdot w=3g$$

where w and r are a function of time so $w(t)$ and $r(t)$.
$g$ is a constant

This equation came from a thought problem of mine which I will now explain since I still do not know how to approach the problem.

Basically I want to design a spiral track in which a vehicle will accelerate. However the total acceleration needs to be constant.

The vehicle will have 2 acceleration components, a radial component ($a_r$) and a change in speed component ($a_s$). Since we want constant acceleration:
$a_r + a_s = constant$

From circular motion we know that $a_r = omega ^2 r$

where $r$ is the radius, and $omega$

Further we know that $v=omega*r$, therefore:
$$a_s=frac{d}{dt}(omega r)$$

For constant acceleration $a_{total} = a_r + a_s = omega ^2 r + frac{d}{dt}(omega r) =constant$

Further usefull information:

Length of the track covered at a certain time
$$l=vcdot t=omega r t$$

With all this the radius will be a function of time, $r(t)$

and the angular velocity will be a function of time, $omega (t)$

Boundary conditions:
$$r(0) = 0$$
$$omega (0) = 0$$

Now I would like to find a possible set of solutions to create a track by implementing end boundary conditions that i choose e.g. $v(t_{end})=100 [m/s]$ and $l(t_{end})=5000 [m]$

Here is a diagram for clarity:
Constant acceleration spiral track

You will not be able to solve this equation since it has two unknowns, you need to impose an additional constraint (and appropriate initial conditions) to solve it. One possible approach would be to choose a function $r=r(t)$, in which case
$$
frac{dv}{dt} = c - frac{v^2} {r}
$$
is the equation to solve, where $c=3g$ frpm the question.

Edit:
If $omega$ is constant then
$$
frac{dr} {dt} = frac{c -omega^2 r} {omega}
$$
which can be solved by seperation of variables.